Component-based reduced-order modeling methods and systems for industrial-scale structural digital twins

Abstract

A method for maintaining physical assets based on recommendations generated from operational data analysis and a composite model representing multiple models of the physical assets includes: constructing the composite model by a computing device using a port-reduced static condensed reduction primitive approximation with respect to at least a portion of a partial differential equation; the computing device analyzing error indices associated with at least one model within the composite model to determine if the error indices exceed a tolerance level, and accordingly increasing the number of basis functions in the port-reduced static condensed reduction primitive approximation; the computing device receiving first operational data associated with at least one region of the physical asset and updating the composite model; and the computing device providing recommendations for maintaining the physical asset based on the updated composite model.

Description

Background Technology

[0001] From a structural integrity perspective, the traditional management approach for industrial machinery and infrastructure involves extensive analysis at the design stage to attempt to assess all relevant operating conditions and, based on this analysis, determine (i) the asset's service life and (ii) a prescriptive scheme for inspection, maintenance, and repair (typically based on fixed time intervals). A key observation in this work is that this design-based prescriptive approach is fundamentally limited by the significant uncertainties inherent in the asset's actual operating conditions. Consider, for example, the case of ocean-going vessels. The vessel is designed based on assumptions about environmental conditions (waves, wind, corrosive seawater) and operating conditions (cargo, the quantity and frequency of loading / unloading cycles). However, future operating conditions are undoubtedly unknown at the design stage; therefore, the only option is to make conservative assumptions and build in large safety margins to compensate for the uncertainty. In practice, relying entirely on this type of design-time analysis can lead to over-design of the asset (corresponding to excessive capital expenditure) or premature retirement compared to its actual structural capacity, or both. Furthermore, even with conservative design assumptions, there remains the risk of unforeseen circumstances exceeding the "worst-case" scenarios assumed during operation, such as extreme weather or accidents. Moreover, the growing trend towards lean design—especially in areas where the economic viability of projects like renewable energy often approaches the break-even point—involves limiting safety margins as much as possible to reduce costs, further increasing the likelihood of assets exceeding their approved operating limits. This introduces health and safety risks and can lead to damage that shortens asset lifespan or requires costly remedial interventions. Therefore, a solution is needed to provide conditional modeling and recommendations for the maintenance and safety of physical assets throughout their entire lifespan. Summary of the Invention

[0002] In one aspect, a method is provided for maintaining physical assets based on recommendations generated from an analytical model of the physical asset, the model comprising multiple components and forming a physics-based digital twin of the physical asset. The method includes constructing a composite model of multiple models using a port-reduced static condensation reduced basis element approximation (PSElement approximation) with respect to at least a portion of partial differential equations. Each of the multiple models represents at least one of the multiple components, and each of the multiple components represents at least one region of the physical asset. The method includes having the computing device analyze an error indicator for at least one of the multiple models to identify error levels associated with the at least one model, in order to determine if the identified error level exceeds a tolerance level. The method includes having the computing device increase the number of basis functions in the port-reduced static condensation reduced basis element approximation based on the determination that the error level of the at least one model exceeds a tolerance level. The method includes repeating the error analysis and the addition of the basis functions for each of the plurality of models until the error level of each of the plurality of models is below a tolerance level. The method includes receiving, by the computing device, first operational data associated with at least one region of the physical asset, represented by at least one parameter of at least one of the plurality of components, from a first operational data source associated with the physical asset. The method includes updating the composite model by the computing device based on the received first operational data. The method includes providing recommendations for maintaining the physical asset by the computing device based on the updated composite model. Attached Figure Description

[0003] The above and other objectives, aspects, features and advantages of this disclosure will become clearer and better understood by referring to the following description taken in conjunction with the accompanying drawings, in which:

[0004] Figure 1A This is an example diagram used to illustrate how to obtain a system-level finite element model by connecting multiple components;

[0005] Figure 1B A hull model updated based on at least one value from the inspection report is described;

[0006] Figure 1CAn updated hull model is described for generating automated instability test reports;

[0007] Figure 2A This is a block diagram describing an embodiment of a system for maintaining physical assets based on recommendations generated from analysis of a physical asset model, the model comprising multiple components and forming a physical-based digital twin of the physical asset;

[0008] Now for reference Figure 2B This figure is a visual block diagram illustrating a method 200 for generating digital threads for floating offshore structures;

[0009] Now for reference Figure 2C The diagram is a visual block diagram illustrating a method 200 for generating digital threads for offshore platforms;

[0010] Figure 3 This is a flowchart describing an embodiment of a method for maintaining physical assets based on recommendations generated from physical asset model analysis, the model comprising multiple components and forming a physical-based digital twin of the physical asset; and

[0011] Figures 4A-4C This is a block diagram illustrating an embodiment of a computer used in conjunction with the methods and systems described herein. Detailed Implementation

[0012] The methods and systems described herein provide functionality for maintaining physical assets based on recommendations generated from physical asset model analysis, wherein the model comprises multiple components and forms a physical-based digital twin of the physical asset. In addition to maintaining physical assets, the methods and systems described herein can provide functionality for identifying one or more aspects of physical assets that should be inspected (e.g., subjected to physical inspection). In addition to maintaining physical assets, the methods and systems described herein can provide functionality for determining the feasibility level of proposed modifications to physical assets. In addition to maintaining physical assets, the methods and systems described herein can provide functionality for determining the operability level of physical assets.

[0013] The methods and systems described herein can provide functionality for performing structural integrity monitoring and reassessment during operation, thereby identifying additional capabilities in an asset (and thus avoiding premature decommissioning) or overly burdensome maintenance regimes, while also tracking the impact of extreme or unpredictable events to ensure safety and reliability. In some embodiments, the goal of asset tracking and integrity monitoring during operation facilitates the concept of a structural digital twin. The term digital twin can refer to a computational replica of a physical asset that is kept in sync with that physical asset throughout its useful life (e.g., based on inspection and sensor data). A structural digital twin can refer to a specific case where the use of a digital twin is to assess structural integrity based on the “current” state of the asset. Updates to a structural digital twin can capture any structurally relevant changes to the asset, and, as an example, can be based on inspection data (e.g., visual inspection, ultrasonic thickness measurement, laser scanning) or sensor measurements (e.g., accelerometers, strain gauges, environmental monitoring). This inspection and instrumentation can significantly reduce (if not eliminate) the uncertainty associated with operating conditions because it allows the digital twin to be continuously updated to reflect the true state of the asset and its environment. Through this approach, users of the methods and systems described herein can develop updated asset management plans (e.g., for inspections, maintenance, repairs, permissible changes in operating conditions, damage or incident response, or asset life extension) informed by the structural digital twin rather than relying on plans developed at design time. Furthermore, it should be noted that the structural digital twin can be provided to physical assets in industrial systems, such as fixed or floating offshore structures, aircraft, mining machinery, rotating machinery, or pressure vessels. Therefore, the methods and systems described herein include a component-based reduced-order modeling framework based on the Static Condensed Reducing Primitives (SCRBE) method, which enables rapid, comprehensive, detailed, and parameterized structural analysis of large-scale industrial systems. This method supports the modeling requirements of structural digital twins, where the structural digital twin is a physically-based detailed model of the structural system that tracks its "current state" throughout the structural system's lifespan. A series of numerical examples will be described in more detail below, demonstrating the unique ability of the SCRBE method to incorporate inspection and / or sensor data during the operation of critical structural systems, effectively perform detailed structural integrity analysis, and enable post-processing and report generation to support data-driven decision-making.

[0014] The methods and systems described herein provide capabilities for generating and updating structural digital twins that satisfy the following four attributes: comprehensive and detailed modeling, speed, parametric modeling, and standards compliance and guaranteed accuracy.

[0015] As an example, a structural digital twin should be able to “screen” the entire system to identify the most likely “failure locations” (e.g., stress and fatigue hotspots), which can then be prioritized in the inspection plan. To achieve this screening, a comprehensive and detailed model is used to accurately address stresses throughout the entire physical asset component. To gain maximum insight from inspection and sensor data, the structural digital twin should represent the entire asset as a holistic model and should provide sufficient detail to accurately map all relevant data to the digital twin. In this way, the digital twin can capture the local, non-local, and cumulative effects of all updates that have been measured or observed (e.g., modifications, defects, damage). This is one aspect of assessing the “current state” of the asset, allowing the system to ensure that the cumulative effects of all updates to date do not lead to new and unintended stress hotspots or predicted failure modes. From a workflow efficiency perspective, comprehensive and detailed modeling is also desirable; in a sense, a detailed model incorporating all the latest asset data provides a “single source of truth” about the asset’s current state and eliminates the need to manage numerous separate, localized models.

[0016] Condition reports from structural digital twins typically involve addressing thousands of different load conditions, such as performing fatigue life estimations on critical components or strength tests according to relevant industry standards in various "hypothetical" scenarios. To enable operators to use digital twins to inform data-driven decision-making, it is necessary to complete all analysis results quickly enough to obtain condition reports in a timely manner for use in the decision-making process. Therefore, the methods and systems described herein provide functionality for generating and updating structural digital twins within a timeframe necessary for the generated or updated structural digital twins to inform decision-making.

[0017] One aspect of structural digital twins is that they can evolve (and in some embodiments, continuously evolve) based on the updated state of the asset or due to operator-imposed modifications to assess different proposed changes or "hypothetical" scenarios. Therefore, the methods and systems described herein provide the capability to generate and update structural digital twins that can be easily and efficiently modified. Parametric modeling methods facilitate such modifications because they allow for alteration of properties, such as stiffness, density, load, geometry, etc., by changing the "dial," and can be automatically and efficiently resolved. Parametric modeling also helps quantify uncertainty because model parameters can be statistically sampled to assess uncertainties in digital twin predictions (e.g., assessing structural fatigue under a range of load scenarios or corrosion rates). This type of statistical uncertainty analysis allows for risk-based asset management programs based on digital twins.

[0018] Structural digital twins are typically developed for safety-critical and high-value assets, and in this regard, much of the existing code from regulatory agencies and standard structures that manage this type of analysis should be used. Therefore, the methods and systems described herein provide functionality for generating and updating structural digital twins that conform to these standards, enabling regulators and operators to fully trust the results they provide while offering analytical findings that can be examined to confirm their accuracy and reliability.

[0019] A common tool for structural integrity analysis of industrial equipment is the finite element method (FE). FE undoubtedly meets the requirements for standard compliance and certifiable accuracy, but it has significant limitations in three other areas. Regarding comprehensive and detailed modeling and speed, FE's computational speed and memory requirements typically grow superlinearly with the number of degrees of freedom, making detailed and comprehensive modeling impossible for most large-scale systems in practical applications. This problem with FE leads to the development of many sub-modeling workflows (coarse global models and separate, fine local models), but sub-modeling methods ignore the nonlocal and cumulative effects we aim to achieve when performing structural digital twin updates. Regarding parametric modeling, FE is not inherently parametric; in a sense, any parameter change requires a new (often computationally intensive) solution to be performed from scratch.

[0020] Artificial intelligence and machine learning (AI / ML), along with related methods such as response surface methodology, are another set of candidate approaches frequently promoted for digital twins. AI / ML enables rapid analysis of systems (typically by evaluating specific quantities of interest (QoIs) according to parameters). Therefore, AI / ML covers project speed and parametric modeling well, but falls short in the other two. For comprehensive and detailed modeling, the evaluation of specific quantities of interest is inconsistent with the concept of a comprehensive and detailed model that should adequately represent all the details of the asset, because a specific QoI output does not provide a picture of the entire asset. As for standards compliance and guaranteed accuracy, AI / ML models are well-known "black boxes" that are difficult to interpret and not based on first principles of physics or meet physics-based standards of asset integrity.

[0021] Several reduced-order modeling (ROM) methods have been developed and can serve as candidate methods for structural digital twins of the types described above. These include parabolic orthogonal decomposition (POD), appropriate generalized decomposition (PGD), or certified reduced-basis methods. ROMs undoubtedly offer speed and—depending on the ROM type—can also provide parametric modeling and guaranteed accuracy. However, ROMs typically cannot provide comprehensive and detailed modeling of large systems. Therefore, they generally do not offer the extensive modeling capabilities required for the largest industrial systems (e.g., large floating structures or aircraft). 8 The above method (FE automation equivalent) is not suitable for large-scale models because it requires solving the full-order model multiple times for different configurations to "train" the ROM, which is too expensive when the full-order model is very large.

[0022] Therefore, the methods and systems described herein provide a component-based ROM approach based on the Static Condensed Reducing Primitives (SCRBE) framework. The SCRBE method provides a physical ROM for parameterized partial differential equations (PDEs) according to a certified basis reduction approach. As is very common for ROM methods, SCRBE involves offline / online decomposition, where model data is “trained” in an offline phase and subsequently evaluated in an online phase for specific parameter choices. The offline phase is computationally intensive, but once that phase is complete, the online phase can be evaluated rapidly for any new parameter choices within a predefined range (typically several orders of magnitude faster than the corresponding FE solution). However, a key aspect that distinguishes this approach from ROM methods as described above is that it is component-based; in a sense, the entire system is decomposed into smaller components, and a separate ROM is trained for each component. This allows for greater scalability compared to other methods because, with SCRBE, the system does not need to use a full-order (e.g., FE) solver to solve the entire system in the offline phase—solving isolated components and local subsystems is sufficient to generate training data. The resulting ROM for each component contains reduced-order representations of the component's internals (using standard basis reduction) and its interfaces (using "port reduction"). The baseline SCRBE method is suitable for linear PDEs because the formula utilizes static condensation, but it can be naturally extended to introduce nonlinearity by including nonlinear FE regions in the model where necessary. This type of SCRBE framework addresses all four properties of structural digital twins as described above.

[0023] While the methods and systems described herein focus on structured digital twins based on the SCRBE approach, in some embodiments these methods and systems may be combined with functionalities for SCRBE, FE, and / or AI / ML. Thus, the methods described herein may include receiving output from an AI / ML system and incorporating that output into the analysis; such methods may include feeding the output back to the AI / ML system, which can then use the output to automatically improve its subsequent execution. In particular, AI / ML works very well as a “canary” because it rapidly generates potential “hazard signals” based on specific QoIs during runtime, which can then be used to apply SCRBE for a comprehensive and detailed analysis to assess hazard signal scenarios in more detail and to prescribe further actions if necessary. Another combination of SCRBE and AI / ML is the use of SCRBE to generate physics-based data that can be used to augment real-world metrics, which can then be used to train a richer AI / ML model. This is crucial for enabling AI / ML models to accurately classify rare behaviors (e.g., failures that are typically rare in well-managed assets), as real-world datasets for rare events are, by definition, finite. To address this issue, physics-based ROMs (such as SCRBE) can be used to efficiently generate diverse fault mode data by simulating specific fault scenarios and extracting virtual sensor readings to augment and enrich the AI / ML training set. Similarly, SCRBE and FE complement each other well because SCRBE can perform fast and parametric modeling of large systems, thereby identifying localized areas within the system where structural integrity issues may exist. Once these areas are identified, extensive local FE analysis can be performed to conduct further evaluations—and because SCRBE is based on an FE mesh, the system can run FE using any subset of components from the SCRBE model.

[0024] While the structural digital twin considered in this work is centered around SCRBE, extensive pre-processing and post-processing layers are still required to provide a complete workflow useful to operators managing critical systems on a daily basis. Such systems should automatically and seamlessly update the structural digital twin based on new data and generate reports summarizing key findings as needed. This process from updated data to the digital twin to new reports and recommendations is often referred to as the “digital thread,” which connects all relevant parts of the digital asset integrity framework. An example of the digital thread—built around an SCRBE-based structural digital twin—will be described in more detail below.

[0025] Component-based modeling has long been the standard method for analyzing large structural systems. The starting point for component-based formulas is typically defining a set of n... comp There are 1 component, where component i corresponds to the spatial domain Ω. i It also includes a set of interfaces—which we call ports—where component i can connect to adjacent components at these interfaces. Assume n port This represents the total number of connected ports in the entire system. As shown in Figure 1, based on this component / port framework, domains are formed by connecting components. An equivalent system-level FE model can be obtained. As shown in Figure 1, components from the hull (top) are assembled into a fully connected system-level model (bottom). The SCRBE method for developing a parameterized ROM, which utilizes the component-based decomposition of the aforementioned system-level model, will be described in more detail below.

[0026] Assumption system:

[0027] KU=F (1)

[0028] This represents the equilibrium structural analysis problem FE proposed on Ω (after applying discretization based on the finite element space), such as static or quasi-static linear elasticity. Here, It is a (symmetric) stiffness matrix. It is a displacement vector, and It is the load vector, where N FE The number of FE degrees of freedom (DOF) in the FE discretization process on Ω is indicated. The system of (1) can be called the standard FE formula for the problem, and in the context of ROM, it is usually referred to as the "truth" formula.

[0029] Regarding the role of components, for ease of illustration, in an n comp =2 and n port In the highly simplified example of N=1, the system-level domain is Ω=Ω1∪Ω2, and p represents a single port connecting Ω1 and Ω2. Assume N FE,1 and N FE,2 These represent the number of FE DOFs associated with the internal (non-port) regions of components 1 and 2, respectively, and assume N. FE,p This indicates the number of FE DOFs on p. It should be noted that the DOFs on p can be standard FE Lagrange base functions associated with individual nodes in the FE mesh, or they can be general functions supporting the entire port—the latter being necessary in the case of “port reduction,” which will be discussed in more detail below. Then, (1) can be rewritten in block form:

[0030]

[0031] The matrix structure here is recommended to follow the following method according to U p Let's solve for U1 and U2:

[0032]

[0033] Substitute (3) into (2), and then generate a result containing only U. p Unknown system:

[0034]

[0035] Or equivalent

[0036]

[0037] The following symbols can be used for the substructure stiffness matrix and load vectors:

[0038]

[0039] in and therefore:

[0040]

[0041] Here, (7) is an exact reconstruction of (1), where the key point is to reduce the system size to N by performing a series of component-local solutions in (3). FE,p ×N FE,p This replaces the initial size N. FE ×N FE This process has many different names in the literature, such as substructuring, superelement, static condensation, or block Gaussian elimination. The name "substructuring" can be used to refer to this method.

[0042] The methods in (2)-(5) depend on the linearity of (1), therefore the component-based formulas discussed in this section are purely linear. Methods applicable to nonlinear analysis will be discussed in more detail below.

[0043] In order to effectively implement (2)-(5), in one embodiment, the calculation is not explicitly performed. Instead, (3) is rewritten as a series of N FE,p +1 solution iterations – each of which corresponds to a DOF on port p, and another corresponds to F. i —As shown below:

[0044]

[0045] Where e j yes The normalized unit vector in the series is 1 in the j-th entry, and once we have completed the series, we can reconstruct... and Therefore, we can use K p,i The terms in (5) are obtained by premultiplication. The X vector in (8) has no physical meaning; its role is only for assembly. and

[0046] In the above derivation, we assume n comp =2 and has only one port p, but this naturally extends to cases with any number of components and ports. Therefore, in the following text, we should refer to N... FE,p This can be understood as the number of FE DOFs on all ports in the system.

[0047] The aforementioned substructured methods have been widely used and offer some attractive computational efficiency and convenience. As mentioned above, one advantage is that the system in (5) is generally much smaller than the system in (1). Another advantage is that if changes are made within component i, then in order to assemble the updated N... FE,p ×N FE,p For the stiffness matrix and load vector, we only need to solve for component i instead of all components (3), thus enabling local modifications to the component-based system in an efficient component-local manner. While these advantages are attractive in some cases, the computational advantages of substructuring compared to the standard FE formula in (1) are generally very limited due to two key issues. First, while the process for introducing changes within the component, as described above, is indeed modular, it can be cumbersome because a new component-local FE solution needs to be performed for each change to the component. In practice, this process can be costly, especially when many updates are required (e.g., when performing modifications to match real-time sensor measurements, or in the inner loop of optimization processes) or when we are using highly analytical component meshes. Second, due to the condensation of the internal DOF, N FE,p ×N FE,p matrix It is indeed less than (1), but generally (in n) compIn the case of >2), it is block sparse, where there may be large and dense blocks of size corresponding to the number of DOFs at the ports on the component interface. Since there is additional density in this matrix structure in many cases of interest, it is possible to require similar or even more computational resources to solve (7) compared to the original sparse FE system (1). This is a well-known problem about substructuring, and the usual advice for solving this problem is to ensure that the ports contain as few DOFs as possible (e.g., by positioning the ports in small or coarse meshed regions) in order to limit the size of the dense blocks. In practice, these requirements impose very strict constraints on the application of substructuring, and in many cases (depending on the model geometry or mesh density), it is impossible to meet these requirements.

[0048] The essence of the problem here is that substructuring is not ROM. Instead, as mentioned above, it is an exact reconstruction of the original FE problem. This ensures complete accuracy in the analysis, but on the other hand, it is limited (potentially severely) by the computational problems described above. Therefore, the methods and systems described here develop a ROM planning based on the substructuring method to address the computational limitations of standard substructuring methods. In particular, the goal here (as with any ROM method) is to introduce small additional approximations compared to non-simplification methods, thereby gaining a significant computational advantage. To address the first problem listed above, in one embodiment, we use a certified basis reduction (RB) method to develop a component-local parameterized ROM so that the FE solution from (3) is replaced with a parameterized RB solution. We introduce parameter vectors It encodes parametric properties for each component, such as stiffness, shell thickness, density, impedance, load, or geometry. Parameter domain μ is defined i The minimum / maximum value of i = 1, ..., D, and according to the RB framework, this field is set before the RB greedy algorithm is executed in the offline phase, because the RB greedy algorithm will... Internal parameters are sampled. Here and in all subsequent development below, we should assume the system under consideration is μ-parameterized. We then create an RB representation for each component so that we can replace the internal FE solution with the parameterized RB solution—as noted in Note 2.1, each port DOF has a solution. The first step in this process is to introduce an affine extension on component i, as follows:

[0049]

[0050] We divide it into parameter-dependent functions (i.e., θ) and parameter-independent operators (i.e., the K matrix and F vector) – as described below, which is crucial for the online efficiency of the RB method. We can then revise the solution sequence from (8) on component i using (9), as follows:

[0051]

[0052] Next, we will define N in (10). FE,p Each of the +1 parameterized equations generates an "RB space". This processing is performed in the "offline" phase using an RB greedy algorithm, which produces a set of decreasing basis functions that accurately represent the entire parameter domain. The complete solution to each equation in (10) above. This RB greedy algorithm achieves this by using residual-based posterior error bounds to guide adaptive sampling in the parameter space, thereby generating an efficient RB model that is accurate across the entire parameter domain of interest.

[0053] This process will result in the generation of N. FE,p +1 RB base. Specifically, assume... Let N represent the “RB basis function matrix” of component i, where N is the basis function matrix of component i. RB,i Z represents the number of RB basis functions, and Z i The j-th column is the j-th basis function. Then, we can “reduce” the parameter-independent operator from (10) as follows:

[0054]

[0055] Furthermore, these reduced operators are stored for later use during online solving. N FE,i and N RB，i Typical sizes can be O(10) 5 O(10) and O(11), so (11) represents the reduction from a large sparse matrix to a very small dense matrix, which can be clearly understood from the RB framework.

[0056] During the “online” phase, to aggregate the contributing factors from within each component, the pre-stored reduction operator from (11) is combined with θ—which is evaluated at the parameter vector μ requested online—to assemble and solve the reduced system for any particular μ of interest. This assembly and solving is fast because it depends only on the size N. RB,i Instead of N FE,i The quantity. This is related to N. FE,iThe online independence is referred to as the offline / online decomposition of the RB method, and it is achieved through an affine extension from (10). In the context of SCRBE, offline / online decomposition allows us to quickly reassemble the system (5) after modifying the parameterized properties inside any component (or all components at the same time), which directly solves the first problem mentioned above.

[0057] Even when exact affine decomposition is unavailable, empirical interpolation (EIM) can be used to achieve an approximate affine decomposition. In practice, this approach is often used to enable certain types of complex parametric processing, including geometric mappings that “deform” the shape of components.

[0058] Generally, in the context of parameterized ROMs, especially in the RB approach, it is well known that the offline and online computational costs of ROMs typically increase rapidly with the number of parameters—this is the so-called "curse of dimensionality." However, the SCRBE framework circumvents this problem through RB greediness and component-based localization of parameters: the RB greedy algorithm on component i only involves a subset of the parameters affecting component i. This means that we can build large systems with many (e.g., thousands) parameters without being affected by the "curse of dimensionality," because the system-level model can be constructed from many components, each with only a few parameters.

[0059] Regarding the second question above, we note that the root of the computational difficulty lies in the fact that traditional substructuring uses N on the port in the notation of (2). FE,p The natural solution to this problem is to reduce the number of DOFs on each port by developing ROMs for the port DOFs. In the SCRBE literature, this process of developing "port ROMs" is called "port reduction" (in some articles, "SCRBE with port reduction" is abbreviated as "PR-SCRBE", but in this paper, we assume that we always apply port reduction, so we always use the shorter name "SCRBE"). We used various port reduction schemes from the literature, such as pairwise training and empirical modes, which involve port-based model simplification with the help of appropriate orthogonal decomposition (POD) or "optimal mode", which obtains a set of optimal port DOFs by solving the transitive feature problem. The goal of these schemes is to construct N DOFs on each port. PR,p (<<N) FE,p The port reduction scheme uses a reduced set of DOFs, while maintaining accuracy compared to the full-order solution by ensuring that the reduced set of port DOFs effectively captures the transmission of dominant information between adjacent components. Furthermore, the port reduction scheme operates offline based on a small sub-model of the entire system, thus eliminating the need for a full-order system-level solution.

[0060] The reduced set of port patterns refers to the fact that we will obtain a size N different from (7). PR,p ×N PR,p The following systems:

[0061]

[0062] Because the size of the dense blocks has been reduced, therefore, Also has less than The density (it should be noted that, as mentioned above, due to the formulation of RB within the component, we now also include parameter dependencies in (12)). Furthermore, since we will include N in (10) FE,p Replaced with N PR,p Therefore, port reduction lowers the offline and online costs associated with the approximation of RBs within the component.

[0063] Port reduction significantly improves the applicability of substructured frameworks because we can use ports of any shape or size and position them anywhere in the model, and as long as we can generate valid ROM for these ports, we can efficiently solve the entire model. In particular, by using SCRBE with port reduction for large models, we typically achieve orders of magnitude speedup compared to standard FE solutions, where, in many cases, as mentioned above, the solution time of a substructured system can be comparable to or even slower than that of standard FE.

[0064] However, compared to traditional FE, the scalability offered by SCRBE with port reduction may be more important than acceleration. The computational cost of using FE to solve large structural models will undoubtedly increase according to N... FEThe growth rate depends on the type of pre-tuner, the type of solver, and the number of conditions for K. In practice, most structural problems related to industrial systems involve one or more forms of ill-conditioning, such as shell components, slender solid components, rigid connectors, or rigid beams, and this ill-conditioning means that iterative solvers with pre-tuning (e.g., pre-tuned conjugate gradient methods, GMRES, or algebraic multigrid methods) are unlikely to converge. Sparse direct solvers are reliable solvers for FE formulas for problem types shown below. One advantage of direct solvers is that they avoid any convergence problems, but a disadvantage is that they introduce significant scalability problems for large-scale problems—especially those related to memory requirements. In contrast, as described below, port-reduced SCRBE can efficiently solve very large-scale models while avoiding convergence problems because—as mentioned above—it does not require performing full-order solutions at the system level in the offline phase, and in the online phase, a system consisting only of port-reduced DOFs small enough to be solved quickly and efficiently by direct solvers can be constructed. Therefore, SCRBE addresses the major computational limitations of traditional FE solvers when solving large-scale structural problems. In summary, the SCRBE framework presented here combines the RB approach regarding component internals with port reduction regarding component interfaces to address both the first and second problems mentioned above. This enables rapid, detailed, and parameterized analysis of large systems, making it a highly suitable set of functionalities for structural digital twins of industrial systems.

[0065] Another category of problems significantly associated with structural digital twins is the frequency domain analysis of forced vibrations, such as Helmholtz acoustics or Helmholtz elastic dynamics. In this case, the FE system takes the following form:

[0066] (K(ν)-ω 2 M(v))U(ω,v)=F(ω,v) (13)

[0067] Where M now represents the discrete mass matrix FE, ω is the frequency, and U is the complex numerical solution vector (e.g., representing pressure (acoustic) or structural response (elastodynamics)). In (13), assuming v represents the user-specified parameter vector (e.g., material properties, geometry, load, impedance), then as described above, the complete parameter vector μ is given by μ = (ω, ν). Assume H(μ) = K(v) - ω 2 M(v), from which we can rewrite (13) as:

[0068] H(μ)U(μ)=F(μ) (14)

[0069] By using the same method as described above, the standard substructured framework (2)-(5) can be applied to (14), and the same drawbacks as the first and second problems exist. Here, the problem of additional density in the matrix structure is as limiting as above, and for the frequency domain, the cost problem of component changes is generally more limiting, since the evaluation of (14) for a single frequency is rarely satisfactory - typically, the goal of frequency domain analysis is to perform a "scan" within the frequency range.

[0070] The SCRBE framework introduced above can be redeployed to address these issues. Doing so brings the same benefits as described above, and also ensures that we can perform efficient frequency scanning using (14), since the frequency ω is naturally incorporated into the SCRBE parameterized ROM—it is simply viewed as an entry of the parameter vector μ.

[0071] The SCRBE framework can be applied almost unchanged to (14). One difference is that we now develop an affine extension of H(μ) instead of K(μ), but this is directly derived from the method introduced above, since the only additional requirement is that we must also create an affine extension of the mass matrix for each component, as follows:

[0072]

[0073] Furthermore, to ensure that all component-internal solutions from (3) are stable, we must impose a constraint on the parameter range of ω, such that ω∈[0,λ]. comp ), where λ comp The smallest eigenvalue across all components in the model of the generalized eigenvalue problem is represented as . in:

[0074]

[0075] Furthermore, we assume that these eigenvalues ​​are ordered, therefore... This constraint guarantees the stability of the local solution of the component, because in λ comp The following Helmholtz equations, isolated for each component, are mandatory, while in λ min As stated above, we face instability because the inf-sup constant decays to zero at resonance. However, it is worth mentioning that λ... comp The maximum value imposed on ω is generally not very restrictive in the context of system-level eigenvalues ​​and eigenmodes, because individual components are usually small compared to the whole system; therefore, λ compThis typically corresponds to high frequencies at the system level. Once an SCRBE model is created and trained for a frequency domain problem, we can usually perform fast “scans” over a wide frequency range and dense sampling in ω to solve complex vibrational responses with high accuracy.

[0076] Next, let's consider the parametric dynamic model:

[0077]

[0078] It represents the equations of motion of the structural system after applying FE discretization in space, where C is the damping matrix, and now the displacement field... It is a function of time. We can then discretize it in time and apply standard explicit or implicit time-marching schemes to solve the system, but this can be a highly computationally intensive approach, especially for large-scale systems. Therefore, a common alternative is to solve the first N eigenvalues ​​of the corresponding eigenvalue problem. modes N eigenvalues / eigenmode pairs (typically, N) modes <<N FE ):

[0079] K(μ)V j (μ)=λ j (μ)M(μ)V j (μ),j=1,…,N modes (18)

[0080] Then, we use modal superposition on a truncated set of eigenmodes to represent the dynamic solution:

[0081]

[0082] in It is a coefficient vector. Equivalently, we can formulate in This indicates that the j-th column is the eigenvector V. j The matrix of (μ). Substituting (19) into (17) and applying the eigenmode orthogonality to M(μ), we obtain:

[0083]

[0084] in It has N modes A system of ODEs—if N modes If it is large enough, then the system can usually capture the global dynamics of the system very well.

[0085] However, if the structural system is large and / or complex, solving the global eigenvalue problem (18) may be computationally infeasible. Therefore, component-based model reduction frameworks have been extensively developed to efficiently compute (18) and (20). The most well-known approach is the Craig-Bampton method, where we form a reduced DOF set for each component based on: (i) interface constraint patterns, which are derived from (8) Same; and (ii) a group A fixed interface normal mode is obtained by leveraging the component-local characteristic problem of imposing zero constraints on all ports. The combination of (i) and (ii) corresponds to using "generalized coordinates". Replace component i As shown below:

[0086]

[0087] in It is the front of component i A matrix of fixed interface normal mode, K i,i (μ) and K p,i (μ) comes from (2), and It is the identity matrix. Because we usually choose... Therefore, since the normal mode of the fixed interface is truncated, this transformation corresponds to a reduction in the number of internal DOFs on component i. This DOF ​​transformation can be applied directly to (17), or it can be applied first to the modal problem (18) and then to the dynamic system as in (20).

[0088] Many extensions to the Craig-Bampton method have been proposed, and they are often grouped into the Component Pattern Synthesis (CMS) family of methods. CMS involves enhancing the Craig-Bampton representation by utilizing additional DOFs about the component interiors. CMS methods (including Craig-Bampton) are effective and widely used component-based ROMs because they capture the main modes and dynamic behavior of the structural system while significantly reducing the number of DOFs by truncating the DOFs within the components. However, we now consider computational considerations regarding CMS, and in particular, we will re-examine the first and second issues above in the context of CMS methods.

[0089] First, due to the extra density in the matrix structure, it is necessary to solve (7) with the same or additional computational resources. In this regard, we note that the standard CMS formulation does not involve port reduction, that is, as mentioned above, I in (21) p,p The block size is N FE,p ×N FE,pHowever, a method has been proposed to address this problem with CMS by reducing the number of port DOFs using truncated eigenmode representations on the ports—although the slow convergence of eigenmode expansion is a limitation of this method, and thus pairwise training, empirical modes, or “optimal modes” methods can offer advantages beyond eigenmode truncation in this context. However, this problem has not been solved by CMS. The dependency on μ in (17)-(21) clearly shows that the parameters required to implement the CMS formulation must be recalculated every time μ is changed. As in the static, quasi-static, and Helmholtz cases, this is a major limitation of CMS for situations where we want to analyze a wide range of model configurations by changing parameters (e.g., in the context of design optimization, or to update real-time models to match sensors or examine data in the context of structural digital twins).

[0090] To address the computational cost issue associated with component changes, a component-based parameterized ROM feature solver based on SCRBE has been previously developed. The core idea of ​​this SCRBE-based feature solver is to reconstruct the feature problem (18) to include the user-specified parameter vector μ and the shift parameter σ, so that:

[0091] (K(μ)-σM(μ))V(μ)=τ(σ,μ)K(μ)V(μ) (22)

[0092] Where τ is the shift eigenvalue, which satisfies:

[0093] τ(λ i (μ),μ)=0 (23)

[0094] Where λ i (μ) comes from (18). The next step is to develop the SCRBE approximation for (22), which follows the same line of reasoning as discussed above for the Helmholtz problem, since the left-hand matrix in (22) is the H matrix from Section 2.2. It should be noted that, as mentioned above, we will restrict σ such that σ∈[0,λ] comp This ensures the coercivity and stability of the internal solution of the component used for H, and due to λ comp This typically corresponds to high frequencies at the system level, so in practice this will again become a moderate limitation.

[0095] Once a SCRBE-based ROM has been constructed for (22), it can be used in an online phase to assemble and solve reduced eigenvalue problems for any value of the pair (σ,μ). In practice, we use this capability as follows: given a user-specified parameter vector μ, we search for values... So that for each j, We will apply iterative root-finding algorithms (such as the Brent method) to find... Then, due to (23), these values ​​will generate the eigenvalues ​​of the original system. Based on this framework, we can use a SCRBE-based ROM to effectively solve the parameterized eigenvalue problem for multiple different values ​​of μ, and according to (20), the generated eigenmodes can also be applied to the dynamic analysis of the parameterized system.

[0096] Next, we consider applying component-based ROMs to nonlinear structural analysis problems. A specific category of nonlinear component-based ROMs is flexible multibody dynamics. This method generalizes from rigid multibody dynamics and is widely used in industrial applications, such as power transmission and robotics modeling, where the system comprises multiple interconnected rigid components. In flexible multibody dynamics, the components in a rigid system can be replaced by ROMs (typically CMS) representing the elastic response of the components. Since the rotation and translation of each component within the multibody system are finite, the overall analysis is geometrically nonlinear, but each flexible component is assumed to have a linear elastic response within its frame of reference. CMS-based flexible multibody dynamics is an effective method within its application domain, but this domain is very specific and cannot meet the range of requirements for structural digital twins, such as detailed stress and fatigue analysis including elastoplasticity, contact / friction, and large strain. Therefore, we focus on other methods here, but we note that since multibody dynamics analysis can provide loading data that can be applied to SCRBE-based models for detailed structural integrity analysis, rigid or flexible body dynamics methods are a natural complement to the model reduction based on (linear or nonlinear) SCRBE discussed here.

[0097] In practice, model reduction for nonlinear systems is often a challenging problem. Various so-called super-reduction strategies have been developed to achieve efficient nonlinear ROMs, such as Empirical Interpolation (EIM), Discrete Empirical Interpolation (DEIM), or the "gappy POD" method. These methods enable complete offline / online decomposition, allowing online ROM assembly and solution to be independent of N. FEAnother approach to nonlinear ROMs is machine learning (ML), where we can non-intrusively train ML models based on supervised learning methods, where a full-order solver provides “real” data. However, each of these approaches inherently comes with computational complexity or accuracy limitations, so in many cases, the computational advantage of ROMs for nonlinear problems can be controversial. This is especially true for “non-smooth” nonlinearities (e.g., contact analysis and elastoplasticity) where small changes in applied loads can cause discrete jumps in the contact surface or the “plastic front”. This type of non-smooth response is inherently difficult to solve for ROMs because ROMs rely on a low-dimensional representation of the response, and if the response is not smooth, then an accurate low-dimensional representation may not exist (mathematically, the Kolmogorov width of the response can be very large, thus preventing effective model reduction). Undoubtedly, many effective and computationally advantageous methods for nonlinear ROM are applicable to specific situations. However, in our view, no single ROM method offers a significant computational advantage over full-order models for the full range of nonlinear analyses related to structural analysis (e.g., contact / friction, elastoplastic and finite strain (e.g. for buckling / post-buckling)).

[0098] Considering the above factors, our approach to obtaining a general nonlinear solver applicable to large-scale structural systems is to apply a so-called hybrid solver, which combines the SCRBE in the linear region and the FE in the nonlinear region in a fully coupled global solution. We first subdivide Ω into two subregions Ω lin and Ω nonlin To formalize it, where Ω nonlin It includes all nonlinearities, and Ω lin It does not contain any nonlinearity.

[0099] In Ω lin Above, we applied the SCRBE framework from Section 2.1, thus giving a size of N. PR,p The reduction system (12). In Ω nonlin Above, we introduced the nonlinear FE operator. in Represents Ω nonlin The number of FE DOFs in the data. Assumptions:

[0100]

[0101] Denotes the global solution of the hybrid SCRBE / FE system, where It is Ω lin The solution above, and It is Ω nonlin The solution on. Generally, regarding Uhybrid This definition allows Ω nonlin With Ω lin There are discontinuities on the interface; therefore, to implement continuity, we introduce a constraint matrix C, which matches the SCRBE port pattern by constraining the FE DOF on the interface, so that the CU hybrid (μ) is continuous. Then, we can write the formula over the entire domain Ω as:

[0102]

[0103] in Let C denote the global nonlinear / linear operator on Ω, and C T The prefactor ensures that we use the same test function as the test function, in the spirit of the Galerkin formula.

[0104] We consider (25) as a nonlinear system with complete bidirectional coupling between the linear and nonlinear regions, and thus we solve it by applying Newton's method to G. The Jacobian matrix of G... It is given by the following formula:

[0105]

[0106] Then, we apply Newton's iteration:

[0107]

[0108]

[0109] Until we reach convergence.

[0110] By using the formulas in (25) and (26) during implementation, we can base it on Ω lin The SCRBE formula and Ω nonlin The FE formula is used to independently assemble the linear and nonlinear parts of the residuals and Jacobian, and then the coupling of these two regions is handled entirely by matrix C.

[0111] In the context of structural digital twins, the hybrid solver approach possesses several attractive properties. First, it offers comprehensive versatility for accurately representing nonlinearities in FE regions. Second, digital twins may require multiple separate nonlinear regions, for example, due to damage, wear, or failure of individual parts of a large system. Due to the fully coupled nature of global nonlinear solutions, the nonlocal and cumulative effects of all these regions are automatically captured by the hybrid solver. (In contrast, the traditional “sub-modeling” workflow ignores nonlocal and cumulative effects). Third, where we have a linear advantage, the hybrid solver offers significant computational benefits compared to full-order solutions, typically a speedup of 100x or more; the linear advantage refers to situations where the number of DOFs in the FE region is significantly less than the number of full-order DOFs in the SCRBE region. We note that the linear advantage is common in structural digital twin applications, where, for example, nonlinearity is typically only required in regions of local damage or failure or in areas of local contact. In cases where we do not have linear advantages (e.g., for large deformation analysis of aircraft wings or wind turbine blades) and the entire model (or almost the entire model) must be treated as nonlinear, a global nonlinear FE solution can be implemented, or one of the other nonlinear RM methods described above can be implemented as appropriate.

[0112] Regarding error metrics and adaptive ROM enrichment methods, posterior error evaluation is a crucial component of the SCRBE framework, used to verify the accuracy of the SCRBE solution in both offline and online phases. This provides guidance on when to stop offline training or when further ROM enrichment is needed in the online phase. Posterior error estimators can be developed for SCRBE methods (using port reduction) related to the formulation of the “true” FE on Ω. The SCRBE error estimator is also developed using a residual-based approach, and to make the method rigorous, many constants are calculated to constrain the error according to the residuals (e.g., those typically required by residual-based error estimators, the stability factor of the operator, and the constants required to constrain the dual norm of the residuals). In the methods and systems described below, we calculate the residuals with respect to the “true” FE space. However, for simplicity, we omit the calculation of the additional constants required for the detailed error estimator and instead use the residuals directly as the posterior error metric. This residual-based error metric is well-suited to the context of structural digital twins because the residuals can be interpreted as a "force balance" criterion, a physically relevant quantity used by structural engineers to indicate the accuracy of the solution. Furthermore, we compute the residuals for discrete full-order systems, a parameter commonly used to determine stopping criteria in the context of iterative solvers (linear Krylov subspace-type methods or nonlinear Newton-type methods). Therefore, the idea of ​​evaluating SCRBE solution accuracy based on this parameter is natural for engineers familiar with iterative solvers.

[0113] To make our residual formula more accurate, we must first introduce It is based on port DOF and is from The coefficients are scaled and the weighted sum of the internal DOFs of the components are used to reconstruct the SCRBE solution over the entire system-level domain Ω. Then, we define the residuals based on (1) as follows.

[0114]

[0115] By separately handling the contributions of the component's internal components and ports to the residuals, the contribution can be evaluated in a computationally efficient manner. Finally, we introduce our error metric. The error metric is the residual norm normalized by the load norm.

[0116] As described below, we use residual-based error metrics in both the offline and online phases. It should be noted that in the following description, we use the concept of a model, which refers to a collection of SCRBE components in which all parameters (materials, geometry, loads, etc.) are specified.

[0117] As already noted, the methods and systems described here, through the use of the SCRBE framework, provide a powerful approach to realizing structural digital twins for large-scale systems. These capabilities are further achieved by connecting SCRBE-based models to inspection and sensor data and configuring post-processing for automated asset integrity reporting. The data flow from running asset data (e.g., sensors or inspections) to structural digital twin updates and analysis, and then to post-processing and reporting, can be referred to as the “digital thread.” This digital thread can provide asset operators with deep structural integrity insights leveraging asset data and SCRBE-based digital twins.

[0118] Now combine Figure 3 Referring to Figure 2, a method 200 is provided for maintaining physical assets based on recommendations generated from an analysis of a physical asset model, the model comprising multiple components and forming a physical-based digital twin of the physical asset. The method includes executing a composite model of multiple models by a computing device using a port-reduced static condensed simplified primitive approximation with respect to at least a portion of partial differential equations, each of the multiple models representing at least one of the multiple components, each of the multiple components representing at least one region of the physical asset (202). The method 200 includes: the computing device analyzing an error index for at least one of the multiple models to identify error levels associated with the at least one model, in order to determine that the identified error level exceeds a tolerance level (204). The method 200 includes: the computing device increasing the number of basis functions in the port-reduced static condensed simplified primitive approximation based on the determination that the error level of the at least one model exceeds a tolerance level (206). The method 200 includes: for each of the plurality of models, repeatedly analyzing the error index and increasing the number of basis functions until the error level of each of the plurality of models is below a tolerance level (208). The method 200 includes: receiving, by the computing device, first operational data associated with at least one region of the physical asset, represented by at least one parameter of at least one of the plurality of components, from a first operational data source associated with the physical asset (210). The method 200 includes: updating the composite model by the computing device based on the received first operational data (212). The method 200 includes: providing recommendations for maintaining the physical asset by the computing device based on the updated composite model (214).

[0119] Therefore, method 200 provides an offline phase that performs multiple steps. Method 200 includes specifying a set of training models. Training tolerance (TOL) and number of training iterations Method 200 includes methods for each model The following steps are performed: (a) solving and calculating the error index ε(μ) using the current SCRBE ROM; and (b) if ε(μ) > TOL, then performing component and port enrichment processing of the SCRBE ROM as described above. Method 200 includes repeating steps (a) and (b). Next, or until for all For ε≤TOL. Method 200 also includes an online phase. In this online phase, for any model M, Method 200 uses the SCRBE ROM generated in the offline phase to solve the system. We can also optionally evaluate ε(μ) to verify the accuracy of the SCRBE solution, and if we find that ε(μ) is greater than expected, we can run further enrichment processing if needed by revisiting the procedure from the offline phase—this processing can be performed fully automatically (driven by an error metric), or the user can choose to guide whether and / or when to perform further enrichment processing. Therefore, the availability of the error metric provides a robust way to ensure the accuracy of our solution in the online phase. Furthermore, if a thorough offline phase is performed, we typically need to rarely revisit the offline computation, and in most cases, we will have a purely online ROM that provides a fast and accurate solution for all systems of interest. We refer to the above offline / online process as Adaptive ROM Enrichment Processing (ARE).

[0120] Now combine Figure 3 For more detailed information, please refer to [link / reference]. Figure 2A A method 200 is provided for maintaining physical assets based on recommendations generated from an analytical model of the physical asset, the model comprising multiple components and forming a physical-based digital twin of the physical asset. The method includes: executing a composite model of multiple models by a computing device using a port-reduced static condensed simplified primitive approximation of at least a portion of partial differential equations, each of the multiple models representing at least one of the multiple components, each of the multiple components representing at least one region of the physical asset (202). The computing device 306 can execute an offline component 308 (which may be provided as a hardware or software component) that uses the SCRBE framework to construct the composite model of the multiple models. As indicated above, in some embodiments a hybrid solver is used to construct a composite model; in such embodiments, at least a first part of the partial differential equation is approximated using the SCRBE method, while FEA is applied to at least a second part of the partial differential equation.

[0121] Method 200 includes the computing device analyzing an error index for at least one of the plurality of models to identify an error level associated with the at least one model, in order to determine that the identified error level exceeds a tolerance level (204). The offline component 308 can solve and calculate the error index using the current SCRBE ROM.

[0122] Method 200 includes increasing the number of basis functions in the port-reduced static condensed simplified primitive approximation by the computing device based on a determination that the error level of the at least one model exceeds a tolerance level (206). If the error index is greater than the error level, then, as described above, offline component 308 enriches the SCRBE ROM (e.g., re-adds it to at least one basis function). Basis functions (or degrees of freedom as described herein) can be added to the component's interface and internally. As described above, the component's internal basis functions can be added after a reduced basis greedy algorithm, where bounds are drawn based on the posterior error of the residuals to guide adaptive sampling in the parameter space, thereby generating an effective RB model that is accurate across the entire parameter domain of interest. The component interface basis functions can be added based on increasing the data used to capture the main information transfer between adjacent components.

[0123] Method 200 includes: for each of the plurality of models, repeatedly analyzing the error index and increasing the number of basis functions until the error level of each of the plurality of models is below a tolerance level (208). In one embodiment, method 200 includes: for each of the plurality of models, repeatedly analyzing the error index and increasing the number of basis functions until the error level of each of the plurality of models is below a tolerance level or until a threshold number of iterations is reached. As an example, method 200 may terminate (208) after determining that each of the plurality of models meets the tolerance level. As another example, method 200 may terminate (208) after a predetermined number of iterations; in this way, if none of the plurality of models meets the tolerance level, method 200 will not continue iterating indefinitely.

[0124] Combined methods 200, (202)-(208) may be referred to as the offline phase. Combined methods 200, (202)-(208) may be performed before generating a visual representation of the composite model. Combined methods 200, (202)-(208) may be performed before the computing device receives first operational data associated with at least one region of the physical asset, represented by at least one parameter of at least one of the multiple components, from a first operational data source associated with the physical asset.

[0125] In some embodiments, before generating an optional visual representation of the composite model or receiving runtime data, method 200 includes receiving user input (e.g., for modeling a "hypothetical" scenario). Therefore, method 200 may include: receiving first user input by a computing device, the first user input identifying an input value indicating at least one physical condition under which a physical asset will be evaluated; and generating at least one output value by the computing device using the composite model, at least in part based on the at least one input value, wherein the at least one output value indicates the behavior of a physical system under the at least one physical condition, wherein the at least one output value comprises multiple output values ​​in an N-dimensional domain. The N-dimensional domain may be a three-dimensional domain or any other value of N. User input may be any of a variety of input types. For example, user input may identify an input value extracted from an inspection report based on an inspection of a physical asset. As another example, user input may identify an input value extracted from runtime data received from sensors associated with the physical asset. As a further example, user input can identify input values ​​used to model components under specific operating conditions, with the purpose of modeling, for example, to perform fatigue life estimation of critical components or to perform strength tests based on industry standards under at least one operating condition.

[0126] In some embodiments, method 200 may include a simulation tool running on a computing device generating a visual representation of the composite model, including visualizing at least one result of a physics-based analysis of the physical asset. In some embodiments, simulation tool 304 generates the visual representation of the composite model. Simulation tool 304 may generate a visual representation of the entire composite model, including visualizing all results of the physics-based analysis of the physical asset. Alternatively, simulation tool 304 may visualize a subset of the obtained values; for example, simulation tool 304 may visualize the stress level at a single weld point instead of the stress level of the entire physical asset. Simulation tool 304 (which may be provided as a hardware or software component) may generate the visual representation. The simulation tool 304 may include a user interface that a user of system 300 can use to interact with the visual representation of the composite model and provide user input. As an example, user interface 314 may allow a user to model the physical system by specifying one or more aspects of the physical system (e.g., geometry, materials, and / or any other suitable physical properties). Once such a model is built, the user can again use the user interface 314 to instruct the simulation tool 304 to perform a simulation based on the model in order to predict how the physical system will behave under one or more selected conditions. The simulation results can be delivered to the user in any suitable manner via the user interface 314, such as by visually presenting one or more output values ​​related to the simulation. Thus, an improved simulation tool is provided that allows the user to modify one or more aspects of the physical system and obtain updated simulation results in real time. For example, by providing the user interface functionality, various modifications can be performed, including but not limited to modifying one or more parameters of a component, adding a component (e.g., by cloning an existing component), removing a component, disconnecting a previously connected component, moving a component from one part of the physical system to another, and rotating a component. In response to such changes requested by the user, the simulation tool is able to quickly provide updated simulation results by utilizing previously calculated data. For example, in some embodiments, the simulation tool can update certain calculations related to components whose parameters and / or connections have changed, but for components not directly affected by these changes, previously calculated data can be reused. According to a further embodiment, the improved simulation tool can perform one or more consistency checks to determine whether the user-requested changes are compatible with other aspects of the physical system. If any incompatibilities are detected, the simulation tool can alert the user. As a supplement or alternative, the simulation tool can propose further changes to the physical system to eliminate one or more incompatibilities introduced by the user-requested changes. According to a further embodiment, an improved simulation tool is provided that automatically calculates the errors associated with the simulation results.In some implementations, the error can be a rigorously calculated error bound, such as the maximum possible difference between the simulation result and the result obtained by calculating the complete FEA solution. For example, in an embodiment that calculates the RB approximation for a component's internal functions, a "local" error bound can be calculated for each bubble function, where the local error bound indicates the difference between the reduced-order model calculated for that bubble function and the corresponding complete FEA solution. Then, by combining such local error bounds, the total error bound for the entire physical system or a portion thereof can be obtained. In other implementations, the error can be an error estimate that can be calculated in a shorter time compared to a rigorous error bound. In still other implementations, the user can choose which type of error (e.g., a rigorous error bound or an error estimate) is calculated by the simulation tool.

[0127] Method 200 includes receiving, by a computing device, first operational data (210) associated with at least one area of ​​a physical asset, represented by at least one parameter of at least one of the plurality of components, from a first operational data source associated with the physical asset. An online component 310 running on computing device 306 may receive the first operational data. Computing device 306 may receive first operational data generated by sensors associated with the physical asset from the first operational data source associated with the physical asset. Computing device 306 may receive first operational data extracted from inspection reports associated with the physical asset from the first operational data source associated with the physical asset; as an example, inspection data may include the results of a visual inspection of the asset (e.g., the detection of cracks or corrosion on pipes, and this information should be built into a digital twin). Computing device 306 may receive first operational data extracted from reports generated by operators of the physical asset from the first operational data source associated with the physical asset. Operational data input to the digital thread may include inspection and sensor data available from the operating asset. Examples include, but are not limited to, thickness measurements based on ultrasonic thickness measurements; environmental monitoring at specific time intervals (e.g., wind and wave conditions for offshore structures); operational load monitoring (e.g., throughput, tank levels, and the number of loading / unloading cycles per time interval); measurements from structural sensors (e.g., accelerometers and strain gauges); and pressure and / or temperature monitoring.

[0128] Method 200 includes updating a composite model (212) by a computing device based on received first operational data. The computing device may identify input values ​​in the received first operational data that indicate at least one physical condition under which a physical asset will be evaluated, and the computing device may also generate at least one output value using the composite model based at least in part on the at least one input value, wherein the at least one output value indicates the behavior of the physical system under the at least one physical condition, and wherein the at least one output value comprises multiple output values ​​over an N-dimensional domain. The computing device may run an importer application to import metric data, which receives the metric data, formats the data, and updates the model; as an example, for thickness measurements, the computing device may receive metric data from a document containing comma-separated value formats (e.g., in a spreadsheet) and update the thickness of the SCRBE model to match the imported metric. For sensor data, the computing device may receive data in an agreed-upon format and import the received data into the SCRBE model. Sensor readings may be received in text format, and the importer software may then be configured to read the agreed-upon text format and apply the data to the model. The computing device can receive identifiers of the sensor's installation location on the physical asset. It can also receive identifiers regarding the format of the inspection data, specifying the location within the physical asset from which each measurement originates. During the initial configuration of the digital twin, the operator or owner of the physical asset and the user generating the digital twin can agree on a formatting process to configure the digital twin in a manner consistent with the operational data to be received. Once the computing device receives operational data from the operational data source, it can use this data as input to the SCRBE model, whether it be operator observations, sensor readings, or inspection reports.

[0129] In some embodiments, simulation tool 304 has already generated a visual representation of the composite model. In such embodiments, simulation tool 304 may update the visual representation of the composite model based on updated output values ​​generated by the composite model using the received first running data.

[0130] Method 200 may include updating the composite model not only once when first running data is received from a first running data source, but also multiple times when multiple running data are received from multiple running data sources. As an example, the digital twin can be continuously updated throughout the online phase to maintain "synchronization" with the physical asset. Method 200 may perform steps for updating the model and any optional generated visual representation based on receiving different data from the same running data source, or based on receiving different data from different running data sources, or both. Therefore, method 200 may include receiving second running data from a first running data source associated with the physical asset by a computing device, the second running data being associated with at least one area of ​​the physical asset represented by at least one parameter of at least one of at least one of a plurality of components; updating the composite model by the computing device based on the received second running data; and providing a second suggestion for maintaining the physical asset by the computing device based on the updated composite model. In an embodiment where simulation tool 304 has already generated a visual representation of the composite model, simulation tool 304 may update the visual representation of the composite model based on updated output values ​​generated by the composite model using the received second running data. Similarly, method 200 may include receiving second operational data from a second operational data source associated with a physical asset by a computing device, the second operational data being associated with at least one second region of the physical asset represented by at least a second parameter of at least a second component of a plurality of components; updating the composite model by the computing device based on the received second operational data; and providing a second recommendation for maintaining the physical asset by the computing device based on the updated composite model. In an embodiment where simulation tool 304 generates a visual representation of the composite model, simulation tool 304 may update the visual representation of the composite model based on updated output values ​​generated by the composite model using the received second operational data.

[0131] As indicated above, method 200 may include periodically updating the offline phase. Updating may include changing values ​​within at least one of a plurality of models. Updating may include replacing at least one of the plurality of models to reflect observed new operating conditions (e.g., by generating inspection reports by operators or by generating sensor data from sensors). Therefore, method 200 may include receiving second operating data from a first operating data source; updating at least one of the plurality of models based on the received second operating data; analyzing, by a computing device, an error index for identifying the error level associated with at least one model to determine whether the identified error level exceeds a tolerance level; increasing the number of basis functions in a port-reduced static condensed simplified primitive approximation based on the determination that the error level of at least one model exceeds the tolerance level; repeating the analysis of the error index and the increase of the number of basis functions for each of the plurality of models until the error level of each of the plurality of models is below the tolerance level; updating the composite model by the computing device based on the received data; and providing recommendations for maintaining physical assets by the computing device based on the updated composite model. In an embodiment where simulation tool 304 has already generated a visual representation of the composite model, simulation tool 304 can update the visual representation of the composite model based on updated output values ​​generated using the received second running data. Similarly, method 200 may include receiving second running data from a second running data source; updating at least one of a plurality of models based on the received second running data; analyzing, by a computing device, an error index for identifying the error level associated with the at least one model to determine whether the identified error level exceeds a tolerance level; increasing the number of basis functions in the port-reduced static condensed simplified primitive approximation based on the determination that the error level of the at least one model exceeds the tolerance level; repeating the analysis of the error index and the increase of the number of basis functions for each of the plurality of models until the error level of each of the plurality of models is below the tolerance level; updating the composite model based on the received data; and providing recommendations for maintaining physical assets based on the updated composite model. In an embodiment where simulation tool 304 has already generated a visual representation of the composite model, simulation tool 304 can update the visual representation of the composite model based on the updated output values ​​generated by the composite model using the received second running data.

[0132] Method 200 includes providing recommendations for maintaining physical assets by a computing device based on an updated composite model (214). System 100 can generate recommendations based on at least one output value calculated through the updated composite model. Typical outputs of digital threads are industry-specific code checks for structural integrity, such as strength, fatigue, and serviceability standards from recognized standards bodies. The calculations required for these standards typically involve post-processing stress data (usually based on hundreds or thousands of different load cases) to calculate parameters such as residual fatigue estimates or buckling utilization.

[0133] Online component 310 can generate recommendations for maintaining physical assets. Online component 310 can provide these recommendations to simulation tool 304 for display to the user via user interface 314. In addition to providing recommendations for maintaining physical assets, as a supplement or alternative, computing device 306 can provide recommendations based on an updated composite model for identifying multiple aspects of the physical assets, in order to examine these aspects in order of priority. In addition to providing recommendations for maintaining physical assets, as a supplement or alternative, computing device 306 can provide recommendations based on an updated composite model for determining the feasibility level of proposed modifications to the physical assets. In addition to providing recommendations for maintaining physical assets, as a supplement or alternative, computing device 306 can provide recommendations based on an updated composite model for determining at least one operating condition of the physical assets.

[0134] User interface 314 may include a dashboard interface where users can click buttons to run tests based on relevant standards and generate reports based on the current state of the digital twin (i.e., incorporating the latest operational data). The report may contain one or more recommendations, such as "Everything is normal," "There is a problem with the XYZ region of the asset," or "The asset cannot be run in the XYZ scene." The output may also be represented as "traffic lights" for each asset, such as "green light" indicating all tests have passed, and "red light" indicating at least one test failed and requiring further attention or investigation by the operator. The operator can then view the full report for more details. In some embodiments, system 100 includes functionality for transmitting the report, including the generated recommendations, to another computing device. In some embodiments, system 100 includes functionality for implementing the recommendations.

[0135] Now for reference Figure 2BThis is a block diagram describing a visual representation of a method 200 for generating digital twins of floating offshore structures. The inputs are sensor data and inspection data, which are used to automatically update and analyze the structural digital twin. The analysis typically involves thousands of SCRBE solutions to assess the asset's strength and fatigue in its "current" state. Once the analysis is complete, a report based on classification society standards is automatically generated.

[0136] Now for reference Figure 2C It is a block diagram describing a visual representation of a method 200 for generating digital threads for offshore platforms. (e.g.) Figure 2C As shown, multiple accelerometers on a structure (e.g., a physical asset) monitor the structure's response to its environment. Method 200 is implemented based on industry standards and generates automated reports.

[0137] As already shown, the SCRBE framework provides four attributes for the aforementioned digital twin modeling. Regarding comprehensive and detailed modeling, the SCRBE framework addresses the issue of high computational cost, thereby enabling efficient solutions with O(10^6) time complexity. 7 ) or O(10 8 A system equivalent to FE DOF. This allows for comprehensive and detailed modeling of large structural systems. Regarding speed, the SCRBE framework meets the requirements due to component-local RB models and port reduction; component-local RBs allow for rapid model modification, and port reduction enables fast system-level solutions. Regarding parametric modeling, component-local RB models conveniently and efficiently introduce parametric ROM functionality, which scales well to a large number of parameters, as each component RB greedy typically only requires handling a few parameters. Regarding standards compliance and certifiable accuracy, SCRBE is essentially a physics-based approach, except for accelerations projected onto a set of reduced port DOFs and component-internal RB DOFs, which are formulated using the same mesh-based approach as FE; this means that all structural integrity analysis standards typically designed for FE are directly applicable to the SCRBE model. Furthermore, as mentioned above, we can evaluate the error estimator or error index of the SCRBE method, thus ensuring the accuracy of any solutions calculated in the online phase. Therefore, the SCRBE-based approach provides all the key functionalities of digital twins to enable the implementation of structural digital twin workflows of interest for large industrial systems.

[0138] The methods and systems described herein provide the functionality for implementing the SCRBE framework, which offers powerful and unique capabilities for structural digital twins of large-scale assets. The adaptive ROM enrichment method can efficiently and reliably train the SCRBE model offline and perform accuracy assessment and enrichment guidance (when needed) online. A series of structural analysis examples demonstrate the scalability, speed, and parameterization capabilities achievable by the SCRBE framework, and the core concepts of the digital thread built around SCRBE-based structural digital twins are described above. The digital thread described here enables an automated framework that provides operators with deeper structural integrity insights based on the "as is" state of critical assets, thereby enabling safer and more efficient operations.

[0139] Figure 1B and 1C Non-limiting examples of the methods and systems described above are provided. See now for reference. Figure 1B It is a block diagram describing the updating of a hull model based on at least one value within an inspection report. This inspection report specifies the thickness of each entity (e.g., plate or stiffener) in the hull, allowing scripts to automatically update the thickness of the corresponding entities in the structural digital twin, thereby incorporating the inspection data. Now refer to... Figure 1C This is a block diagram describing an updated hull model used to generate automated buckling test reports. The left diagram shows each entity in the hull (e.g., plates and stiffeners). The lower middle diagram shows stress data extracted for each entity, and then the buckling criterion specifies a formula used to evaluate each entity to provide a utilization value. In the upper middle diagram, we display the utilization value as a "heat map" of the hull, and any entity with a utilization value exceeding 1 is considered a failure. The system then generates a report identifying any entities that failed the buckling test (right diagram). Operators of physical assets can use such reports to prioritize inspections, maintenance, and repairs, or to assess the structural health of the asset.

[0140] Now for reference Figure 4A , 4B And 4C, these are block diagrams describing additional details of computing devices that can be modified to perform functions used to implement the methods and systems described above. Now refer to Figure 4AThe figure illustrates an embodiment of a network environment. In short, the network environment includes one or more clients 102a-102n (also commonly referred to as one or more local machines 102, one or more clients 102, one or more client nodes 102, one or more client machines 102, one or more client computers 102, one or more client devices 102, one or more computing devices 102, one or more endpoints 102, or one or more endpoint nodes 102) communicating via one or more networks 404 with one or more remote machines 106a-106n (also commonly referred to as one or more servers 106 or one or more computing devices 106).

[0141] Although Figure 4A A network 404 is shown between one or more clients 102 and a remote machine 106, but the one or more clients 102 and the remote machine 106 may reside on the same network 404. Network 404 may be a local area network (LAN) (e.g., a corporate intranet), a metropolitan area network (MAN), or a wide area network (WAN) (e.g., the Internet or the World Wide Web). In some embodiments, multiple networks 404 exist between one or more clients and the remote machine 106. In one embodiment, network 404' (not shown) may be a private network, and network 404 may be a public network. In another embodiment, network 404 may be a private network and network 404' may be a public network. In yet another embodiment, both networks 404 and 404' may be private networks. In yet another embodiment, both networks 404 and 404' may be public networks.

[0142] Network 404 can be any type and / or form of network, and can include any of the following: point-to-point network, broadcast network, wide area network, local area network, telecommunications network, data communication network, computer network, ATM (Asynchronous Transfer Mode) network, SONET (Synchronous Optical Network) network, Synchronous Digital Hierarchy (SDH) network, wireless network, and wired network. In some embodiments, network 404 may include a wireless link, such as an infrared channel or satellite band. The topology of network 404 can be a bus, star, or ring network topology. Network 404 can be any such network topology known to those skilled in the art capable of supporting the operations described herein. Network 404 may include a mobile phone network using any one or more protocols for communication between mobile devices (typically including tablets and handheld devices), including AMPS, TDMA, CDMA, GSM, GPRS, UMTS, or LTE. In some embodiments, different types of data can be transmitted using different protocols. In other embodiments, the same type of data can be transmitted using different protocols.

[0143] One or more client 102 and remote machine 106 (generally referred to as computing device 100) can be any workstation, desktop computer, laptop or notebook computer, server, portable computer, mobile phone, mobile smartphone or other portable telecommunications device, media playback device, gaming system, mobile computing device, or any other type and / or form of computing, telecommunications or media device capable of communicating over any type and form of network and having sufficient processor power and memory capacity to perform the operations described herein. Client 102 can run, operate or otherwise provide an application, which can be any type and / or form of software, program or executable instructions, including but not limited to any type and / or form of web browser, web-based client, client-server application, ActiveX control or JAVA applet, or any other type and / or form of executable instructions capable of running on client 102.

[0144] In one embodiment, computing device 106 provides web server functionality. In some embodiments, web server 106 includes an open-source web server, such as the NGINX web server provided by NGINX, Inc. in San Francisco, California, or the Apache Apache server maintained by the Apache Software Foundation in Delaware. In other embodiments, the web server executes proprietary software, such as the INTERNET INFORMATION SERVICES product provided by Microsoft Corporation in Redmond, Washington, the ORACLE IPLANET web server product provided by Oracle Corporation in Redwood Beach, California, or the BEAWEBLOGIC product provided by BEA Systems in Santa Clara, California.

[0145] In some embodiments, the system may include multiple logically grouped remote machines 106. In one embodiment, the logical group of remote machines may be referred to as server group 438. In another embodiment, server group 438 may be managed as a single entity.

[0146] Figure 4B and 4C A block diagram of a computing device 100 is described for implementing one or more client 102 or remote machine 106 embodiments. Figure 4B and 4C As shown, each computing device 100 includes a central processing unit 421 and a main memory unit 422. For example... Figure 4B As shown, computing device 100 may include storage device 428, mounting device 416, network interface 418, I / O controller 423, display device 424a-n, keyboard 426, pointing device 427 (e.g., mouse), and one or more other I / O devices 430a-n. Storage device 428 may include, but is not limited to, operating system and software. Figure 4C As shown, each computing device 100 may also include additional optional components that communicate with the central processing unit 421, such as a memory port 403, a bridge 470, one or more input / output devices 430a-n (generally referred to by reference numeral 430), and a cache memory 440.

[0147] Central processing unit 421 is any logic circuit that responds to and processes instructions fetched from main memory unit 422. In many embodiments, central processing unit 421 is provided by a microprocessor unit, such as those manufactured by companies including: Intel Corporation of Mountain View, California; Motorola Corporation of Schaumburg, Illinois; Transmeta Corporation of Santa Clara, California; International Business Machines of White Plains, New York; or Advanced MicroDevices of Sunnyvale, California. Other examples include SPARC processors, ARM processors, processors for building UNIX / LINUX “white” boxes, and processors for mobile devices. Computing device 400 may be based on any of these processors, or on any other processor capable of operating in the manner described herein.

[0148] Main memory unit 422 can be one or more memory chips capable of storing data and allowing microprocessor 421 direct access to any memory location. Main memory 422 can be based on any available memory chip capable of operating in the manner described herein. Figure 4B In the illustrated embodiment, the processor 421 communicates with the main memory 422 via the system bus 450. Figure 4C An embodiment of computing device 400 is described, wherein the processor communicates directly with main memory 422 via memory port 403. Figure 4C An embodiment in which the main processor 321 communicates directly with the cache memory 440 via an auxiliary bus (sometimes referred to as a back-end bus) is also described. In other embodiments, the main processor 421 uses the system bus 450 to communicate with the cache memory 440.

[0149] exist Figure 4B In the illustrated embodiment, processor 421 communicates with various I / O devices 430 via a local system bus 450. Different buses can be used to connect central processing unit 421 to any I / O device 430, including VESAVL bus, ISA bus, EISA bus, Micro Channel Architecture (MCA) bus, PCI bus, PCI-X bus, PCI-Express bus, or NuBus. In embodiments where the I / O device is a video display 424, processor 421 can use an Advanced Graphics Port (AGP) to communicate with display 424. Figure 4CAn embodiment of computer 400 is described, wherein the main processor 421 also communicates directly with I / O device 430b via communication technologies such as HYPER TRANSPORT, RAPIDIO, or INFINIBAND.

[0150] One or more of various I / O devices 430a-n may exist within or be connected to the computing device 400, wherein each I / O device may be of the same or different type and / or form. Input devices include keyboard, mouse, touchpad, trackball, microphone, scanner, camera, and drawing tablet. Output devices include video display, speaker, inkjet printer, laser printer, 3D printer, and dye-sublimation printer. I / O devices may be composed of, for example... Figure 4B The I / O controller 423 shown controls the device. Additionally, the I / O device may provide storage and / or mounting media 416 for the computing device 400. In some embodiments, the computing device 400 may provide a USB connection (not shown) to receive a handheld USB storage device (e.g., a series of USB flash drives manufactured by Twintech Industry, Inc., Los Alamitus, California).

[0151] Still referencing Figure 4B The computing device 100 may support any suitable installation device 416, such as a floppy disk drive for receiving floppy disks (e.g., 3.5-inch, 5.25-inch disks, or ZIP disks); a CD-ROM drive; a CD-R / RW drive; a DVD-ROM drive; tape drives of various formats; a USB device; a hard disk drive; or any other device suitable for installing software and programs. In some embodiments, the computing device 400 may provide functionality for installing software via a network 404. The computing device 400 may also include storage devices for storing the operating system and other software, such as a redundant array of one or more hard disks or individual disks. Alternatively, the computing device 100 may rely on memory chips instead of hard disks for storage.

[0152] Furthermore, computing device 400 may include a network interface 418 for connection to network 404 via various connections, including but not limited to standard telephone lines, LAN or WAN links (e.g., 802.11, T1, T3, 56kb, X.25, SNA, DECNET), broadband connections (e.g., ISDN, Frame Relay, ATM, Gigabit Ethernet, SONET-based Ethernet), wireless connections, or some combination of any or all of the above. Connections can be established using various communication protocols (e.g., TCP / IP, IPX, SPX, NetBIOS, Ethernet, ARCNET, SONET, SDH, Fiber Distributed Data Interface (FDDI), RS232, IEEE 802.11, IEEE 802.11a, IEEE 802.11b, IEEE 802.11g, IEEE 802.11n, 802.15.4, Bluetooth, ZigBee, CDMA, GSM, WiMax, and direct asynchronous connections). In one embodiment, computing device 400 communicates with other computing devices 100' via any type and / or form of gateway or tunneling protocol (e.g., Secure Sockets Layer (SSL) or Transport Layer Security (TLS)). Network interface 418 may include a built-in network adapter, network interface card, PCMCIA network card, bus network adapter, wireless network adapter, USB network adapter, modem, or any other device suitable for connecting computing device 100 to any type of network capable of communication and suitable for performing the operations described herein.

[0153] In a further embodiment, the I / O device 430 may be a bridge between the system bus 150 and an external communication bus, such as a USB bus, Apple Desktop Bus, RS-232 serial connection, SCSI bus, FireWire bus, FireWire 800 bus, Ethernet bus, AppleTalk bus, Gigabit Ethernet bus, Asynchronous Transfer Mode bus, HIPPI bus, Super HIPPI bus, SerialPlus bus, SCI / LAMP bus, FibreChannel bus, or a serial connection small computer system interface bus.

[0154] Figure 4B and 4CThe computing device 400 of the type described herein typically operates under the control of an operating system, which controls task scheduling and access to system resources. The computing device 400 can run any operating system, such as any version of Microsoft Windows, different versions of UNIX and Linux, any version of MAC OS for Macintosh computers, any embedded operating system, any real-time operating system, any open-source operating system, any proprietary operating system, any operating system for mobile computing devices, or any other operating system capable of running on a computing device and performing the operations described herein. Typical operating systems include, but are not limited to: Windows 3.x, Windows 95, Windows 98, Windows 2000, Windows NT 3.1-4.0, Windows CE, Windows XP, Windows 7, Windows 8, Windows Vista, and Windows 10 manufactured by Microsoft Corporation of Redmond, Washington; any version of MAC OS manufactured by Apple Inc. of Cupertino, California; OS / 2 manufactured by International Business Machines of Armonk, New York; Red Hat Enterprise Linux, a variant of the Linus OS distributed by Red Hat, Inc. of Raleigh, North Carolina; Ubuntu, a free operating system distributed by Canonical Ltd. of London, UK; or any other type and / or form of Unix operating system, etc.

[0155] Computing device 400 can be any workstation, desktop computer, laptop or notebook computer, server, portable computer, mobile phone or other portable telecommunications device, media playback device, gaming system, mobile computing device, or any other type and / or form of computing, telecommunications, or media device capable of communication and having sufficient processor power and storage capacity to perform the operations described herein. In some embodiments, computing device 100 may have a different processor, operating system, and input device compatible with the device. In other embodiments, computing device 400 is a mobile device, such as a JAVA-enabled cellular phone / smartphone or personal digital assistant (PDA). The computing device 400 may be a mobile device, such as, but not limited to, mobile devices manufactured by the following companies: Apple Inc. of Cupertino, California; Google / Motorola Div. of Fort Worth, Texas; Kyocera of Kyoto, Japan; Samsung Electronics Co., Ltd. of Seoul, South Korea; Nokia of Finland; Hewlett-Packard Development Company, LP and / or Palm, Inc. of Sunnyvale, California; Sony Ericsson Mobile Communications AB of Lund, Sweden; or Research In Motion Limited of Waterloo, Ontario, Canada. In other embodiments, the computing device 100 may be a smartphone, a PCKET PC, a PCKET PCPHONE, or other portable mobile device that supports Microsoft Windows Mobile software.

[0156] In some embodiments, computing device 400 is a digital audio player. In one embodiment, computing device 400 is a digital audio player, such as the Apple iPod, iPod Touch, iPod Nano, and iPod SHUFFLE device series manufactured by Apple Inc. In another embodiment, the digital audio player may function as both a portable media player and a mass storage device. In other embodiments, computing device 400 is a digital audio player, such as, but not limited to, digital audio players manufactured by companies like Samsung Electronics America of Ridgefield Park, New Jersey, or Creative Technologies Ltd. of Singapore. In other embodiments, computing device 400 is a portable media player or digital audio player that supports file formats including, but not limited to, MP3, WAV, M4A / AAC, WMA, protected AAC, AEFF, audiobooks, Apple Lossless Audio File Format, .mov, .m4v, and .mp4 MPEG-4 (H.264 / MPEG-4 AVC) video file formats.

[0157] In some embodiments, computing device 400 includes a device combination, such as a mobile phone combined with a digital audio player or portable media player. In one embodiment, computing device 100 is a device in a Google / Motorola series of combined digital audio players and mobile phones. In another embodiment, computing device 400 is a device in the iPhone smartphone series manufactured by Apple Inc. In yet another embodiment, computing device 400 is a device running the Android open-source mobile phone platform distributed by the Open Handset Alliance; for example, device 100 could be a device provided by Samsung Electronics in Seoul, South Korea, or HTC headquarters in Taiwan, China. In other embodiments, computing device 400 is a tablet device, such as, but not limited to, iPad series devices manufactured by Apple Inc.; Playbook manufactured by Research In Motion; CRUZ series devices manufactured by Velocity Micro, Inc. of Richmond, Virginia; FOLIO and THRIVE series devices manufactured by Toshiba America Information Systems, Inc. of Irvine, California; GALAXY series devices manufactured by Samsung; HPSLATE series devices manufactured by Hewlett-Packard; and STREAK series devices manufactured by Dell, Inc. of Round Rock, Texas.

[0158] Computing device 400 may be a file server, application server, web server, proxy server, appliance, network appliance, gateway, application gateway, gateway server, virtualization server, deployment server, SSL VPN server, or firewall. In some embodiments, computing device 400 provides remote authentication dial-up user services and is referred to as a RADIUS server. In other embodiments, computing device 400 may have the capability to function as an application server or a main application server. In other embodiments, computing device 400 is a blade server.

[0159] The computing device 400 may be referred to as a client node, client machine, endpoint node, or endpoint. In some embodiments, the client 400 has the capability to act both as a client node seeking access to resources provided by the server and as a server node providing access to hosted resources to other clients.

[0160] In some embodiments, a first client computing device 400a communicates with a second server computing device 400b. In one embodiment, the client communicates with one of the computing devices 400 in a server cluster. For example, via a network, the client can request execution of various applications hosted by the computing devices 400 in the server cluster and can receive and display the output data of the application execution results.

[0161] Some embodiments of methods and systems for maintaining physical assets based on recommendations generated from physical asset model analysis have already been described herein, the models comprising multiple components and forming a physical-based digital twin of the physical assets. It will now be apparent to those skilled in the art that other embodiments incorporating the concepts of this disclosure are also applicable.

Claims

1. A method for maintaining physical assets based on recommendations generated from an analytical physical asset model, said model comprising multiple components and forming a physical-based digital twin of said physical assets, said method comprising: (a) A composite model of multiple models constructed by a computing device using a port-reduced static condensed simplified primitive approximation of at least a portion of the partial differential equations, each of the multiple models representing at least one of the multiple components, each of the multiple components representing at least one region of the physical asset. (b) The computing device analyzes an error index used to identify the error level associated with the at least one of the plurality of models in order to determine whether the identified error level exceeds the tolerance level; (c) The computing device increases the number of basis functions in the port-reduced static condensed simplified primitive approximation based on a determination by the computing device that the error level of the at least one model exceeds the tolerance level; (d) For each of the plurality of models, repeat (b) and (c) until the error level of each of the plurality of models is lower than the tolerance level; (e) The computing device receives first operational data associated with at least one region of the physical asset, represented by at least one parameter of at least one of the plurality of components, from a first operational data source associated with the physical asset, wherein the first operational data is received from a sensor associated with the physical asset. (f) The computing device updates the composite model based on the received first operational data; and (g) The computing device provides recommendations for maintaining the physical asset based on an updated composite model, wherein the recommendations identify multiple aspects of the physical asset to be examined, the multiple aspects being arranged according to priority levels based on the updated composite model.

2. The method of claim 1, wherein (a)-(d) are performed before (e).

3. The method of claim 1, further comprising: Following (a)-(d) and preceding (e), the composite model is used to generate a physical-based analysis of the physical assets, wherein the generation further includes: The computing device receives first user input, which identifies an input value indicating at least one physical condition under which the physical asset will be evaluated. The computing device generates at least one output value using the composite model based at least in part on the at least one input value, wherein the at least one output value indicates the behavior of the physical system under the at least one physical condition, and wherein the at least one output value includes a plurality of output values ​​in an N-dimensional domain.

4. The method of claim 3, wherein receiving further comprises: The computing device receives user input for identifying input values ​​extracted from an inspection report, wherein the inspection report is based on a physical inspection of the physical asset.

5. The method of claim 3, wherein receiving further comprises: The computing device receives user input for identifying input values ​​extracted from operational data, wherein the operational data is received from sensors associated with the physical asset.

6. The method of claim 1, further comprising: (h) A simulation tool running on the computing device generates a visual representation of the composite model, wherein the visual representation includes a visual representation of at least one result of a physical-based analysis of the physical asset.

7. The method of claim 6, further comprising: (i) The simulation tool updates the visual presentation based on the first running data received.

8. The method of claim 1, wherein (d) comprises: Repeat (b) and (c) for each of the multiple models until the threshold number of iterations is reached.

9. The method of claim 1, wherein (e) further comprises: The computing device receives first operational data generated by sensors associated with the physical asset from the first operational data source associated with the physical asset.

10. The method of claim 1, wherein (e) further comprises: The computing device receives first operational data extracted from inspection reports associated with the physical asset from the first operational data source associated with the physical asset.

11. The method of claim 1, wherein (e) further comprises: The computing device receives first operational data extracted from reports generated by operators of the physical asset from the first operational data source associated with the physical asset.

12. The method of claim 1, further comprising: (h) The computing device provides recommendations for determining the feasibility level of proposed modifications to the physical asset and / or for determining the operability level of the physical asset, based on an updated composite model.

13. The method of claim 1, further comprising: (h) The computing device receives second operational data associated with at least one region of the physical asset, represented by at least one parameter of at least one of the plurality of components, from the first operational data source associated with the physical asset; (i) The computing device updates the composite model based on the received second operational data; as well as (j) The computing device provides a second recommendation for maintaining the physical assets based on an updated composite model.

14. The method of claim 1, further comprising: (h) The computing device receives second operational data associated with at least a second region of the physical asset, represented by at least a second parameter of at least a second component of the plurality of components, from a second operational data source associated with the physical asset; (i) The computing device updates the composite model based on the received second operational data; as well as (j) The computing device provides a second recommendation for maintaining the physical assets based on an updated composite model.

15. The method of claim 1, further comprising: (h) Receive second running data from the first running data source; (i) Update at least one of the plurality of models based on the received second running data; (j) For at least one of the plurality of models, the computing device analyzes an error index used to identify the error level associated with the at least one model in order to determine whether the identified error level exceeds the tolerance level; (k) Based on the determination that the error level of the at least one model exceeds the tolerance level, the computing device increases the number of basis functions in the port-reduced static condensed simplified primitive approximation; (l) Repeat (b) and (c) for each of the plurality of models until the error level of each of the plurality of models is lower than the tolerance level; (m) The computing device updates the composite model based on the received second operational data; as well as (n) The computing device provides recommendations for maintaining the physical assets based on an updated composite model.

16. The method of claim 1, further comprising: (h) Receive second running data from the second running data source; (i) Update at least one of the plurality of models based on the received second running data; (j) For at least one of the plurality of models, the computing device analyzes an error index used to identify the error level associated with the at least one model in order to determine whether the identified error level exceeds the tolerance level; (k) Based on the determination that the error level of the at least one model exceeds the tolerance level, the computing device increases the number of basis functions in the port-reduced static condensed simplified primitive approximation; (l) Repeat (b) and (c) for each of the plurality of models until the error level of each of the plurality of models is lower than the tolerance level; (m) The computing device updates the composite model based on the received second operational data; as well as (n) The computing device provides recommendations for maintaining the physical assets based on an updated composite model.

17. A non-transitory computer-readable medium encoded with computer-executable instructions, wherein when run on a computing device, the computer-executable instructions cause the computing device to perform the method according to any one of claims 1 to 16.

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