Multi-module date-driven order reduction method for static and vibrational structural analysis of turbine blades

Abstract

A multimodule data-based order reduction method for static and vibration structural analysis of a turbine blade according to one embodiment may include a preprocessing operation of receiving a discretized model and boundary conditions based on a geometric shape model of a turbine blade, an operation of analyzing an overall order model based on information and parameters input from the preprocessing operation, an operation of collecting the results of the overall order model analysis and building a database, an operation of extracting a suitable orthogonal decomposition mode by performing singular value decomposition on the built database, and an operation of generating an order reduction model.

Description

Technology Field

[0001] The present invention relates to a multimodal data-based order reduction method for static and vibrational structural analysis of turbine blades. Background Technology

[0002] Gas turbines are high-efficiency thermal engines utilized in various industrial sectors, including power generation, aviation, and marine. The turbine blades are a critical component that determines the engine's performance and efficiency. During operation, turbine blades are exposed to complex load conditions, such as high-temperature, high-pressure air excitation forces and centrifugal forces resulting from high-speed rotation. Faced with extreme environments, blades face an increased risk of deformation and damage; therefore, ensuring structural integrity through forced vibration and structural response analysis during the design phase is essential.

[0003] To execute the design, the stress and deformation of the blade must be precisely evaluated based on parameters such as material properties and rotational speed. In particular, high-precision structural analysis is essential to predict stress concentration sites that may occur under complex geometries and various loading conditions, and to accurately evaluate the resulting structural behavior.

[0004] The finite element method (FEM) is primarily utilized for structural analysis. The FEM is a technique that derives the behavior of a structure by discretizing it into a finite number of elements and numerically analyzing the equilibrium equations of each element, requiring large-scale matrix operations.

[0005] In particular, large-scale discretization is essential for turbine blades to accurately reflect numerous curved shapes, such as cooling holes and root joints. Furthermore, high-dimensional matrix operations are required to precisely simulate physical phenomena like spin-softening and stress-stiffening caused by centrifugal force, as well as nonlinear behaviors such as large deformations. High-precision analysis considering these requirements demands excessive computational resources and time, and there are limitations in rapidly reviewing various design options based on parameter changes during the design phase. means of solving the problem

[0006] A multimodule data-based order reduction method for static and vibration structural analysis of a turbine blade according to one embodiment may include a preprocessing operation of receiving a discretized model and boundary conditions based on a geometric shape model of a turbine blade, an operation of analyzing an overall order model based on information and parameters input from the preprocessing operation, an operation of collecting the results of the overall order model analysis and building a database, an operation of extracting a suitable orthogonal decomposition mode by performing singular value decomposition on the built database, and an operation of generating an order reduction model.

[0007] In one embodiment, the operation of analyzing the entire order model may include linear and nonlinear static analysis, time transient analysis, and forced vibration analysis.

[0008] In one embodiment, the parameter may relate to the material properties or operating conditions of the turbine blade.

[0009] In one embodiment, the material properties of the turbine blade may be the density or elastic modulus of the turbine blade.

[0010] In one embodiment, the operating condition of the turbine blade may be the rotational speed of the turbine blade or the magnitude of the applied load.

[0011] In one embodiment, the operation of generating the order reduction model can be performed using a Galerkin projection method.

[0012] An electronic device for static and vibration structural analysis of a turbine blade according to one embodiment includes at least one processor and a memory for storing instructions, and when the instructions are executed individually or collectively by the at least one processor, the electronic device may be configured to receive a discretized model and boundary conditions based on a geometric model for a designed turbine blade, analyze a full-order model based on information and parameters input regarding the discretized model and boundary conditions, collect the results of the full-order model analysis to build a database, perform singular value decomposition on the built database to extract a suitable orthogonal decomposition mode, and generate a reduced-order model. Brief explanation of the drawing

[0013] FIG. 1 is a flowchart of a multimodal data-based order reduction method for static and vibration structural analysis of a turbine blade according to one embodiment. FIG. 2 is a block diagram of an electronic device according to one embodiment. FIG. 3 is a perspective view illustrating a turbine blade to which a structural analysis method according to one embodiment is applied. FIG. 4 is a block diagram schematically illustrating parameter conditions according to one embodiment and a multimodal data-based order reduction process according thereto. Figure 5 is a perspective view illustrating the boundary conditions of an analysis reflecting the parameter conditions of Figure 4. FIG. 6 is a diagram schematically illustrating the process of storing static analysis, time transient analysis, and forced vibration analysis results according to parameter conditions in one embodiment. FIG. 7 is a diagram schematically illustrating a database and POD basis extraction process according to one embodiment. FIG. 8 is a diagram showing the deformation distribution of the full order model and the order reduction model according to one embodiment. FIG. 9 is a graph showing the maximum displacement of the turbine blade tip over time in each of the full order model and the reduced order model according to one embodiment. Specific details for implementing the invention

[0014] Specific structural or functional descriptions of the embodiments are disclosed for illustrative purposes only and may be modified and implemented in various forms. Accordingly, actual implementations are not limited to the specific embodiments disclosed, and the scope of this specification includes modifications, equivalents, or substitutions included in the technical concept described by the embodiments.

[0015] Terms such as first or second may be used to describe various components, but these terms should be interpreted solely for the purpose of distinguishing one component from another. For example, the first component may be named the second component, and similarly, the second component may also be named the first component.

[0016] When it is stated that a component is "connected" to another component, it should be understood that it may be directly connected to or joined to that other component, or that there may be other components in between.

[0017] The singular expression includes the plural expression unless the context clearly indicates otherwise. In this specification, terms such as "comprising" or "having" are intended to specify the existence of the described features, numbers, steps, actions, components, parts, or combinations thereof, and should be understood as not precluding the existence or addition of one or more other features, numbers, steps, actions, components, parts, or combinations thereof.

[0018] Components included in any one embodiment and components having common functions shall be described using the same names in other embodiments. Unless otherwise stated, the descriptions given in any one embodiment may also apply to other embodiments, and specific descriptions shall be omitted to the extent of overlap.

[0019] Unless otherwise defined, all terms used herein, including technical or scientific terms, have the same meaning as generally understood by those skilled in the art. Terms such as those defined in commonly used dictionaries should be interpreted as having a meaning consistent with their meaning in the context of the relevant technology, and should not be interpreted in an ideal or overly formal sense unless explicitly defined in this specification.

[0020] Hereinafter, embodiments will be described in detail with reference to the attached drawings. In the description with reference to the attached drawings, identical components are given the same reference numeral regardless of the drawing number, and redundant descriptions thereof will be omitted.

[0021] FIG. 1 is a flowchart of a multimodal data-based order reduction method for static and vibration structural analysis of a turbine blade according to one embodiment, FIG. 2 is a block diagram of an electronic device according to one embodiment, FIG. 3 is a perspective view showing a turbine blade to which a structural analysis method according to one embodiment is applied, FIG. 4 is a block diagram schematically showing parameter conditions and a multimodal data-based order reduction process according to one embodiment, FIG. 5 is a perspective view showing boundary conditions of analysis reflecting the parameter conditions of FIG. 4, FIG. 6 is a diagram schematically showing the process of storing static analysis, time transient analysis, and forced vibration analysis results according to parameter conditions according to one embodiment, and FIG. 7 is a diagram schematically showing the database and POD basis extraction process according to one embodiment.

[0022] Referring to FIGS. 1 to 7, a multimodal data-based order reduction method for static and vibration structural analysis of a turbine blade (9) according to one embodiment can predict structural behavior more accurately and efficiently by reducing the order when performing structural analysis required in the design stage of the turbine blade (9). To this end, the present invention uses a reduced order model (ROM) based on singular value decomposition (SVD) and proper orthogonal decomposition (POD) instead of the finite element method using a conventional full order model (FOM).

[0023] In one embodiment, a multimodal data-based order reduction method may be performed through an electronic device (2). The electronic device (2) may include a control unit (21) capable of performing structural analysis of a turbine blade (9) and applying an order reduction model, an input unit (22) for inputting initial conditions for performing multimodal data-based order reduction or analysis parameters of the order reduction model to the control unit (21), and a display (23) for outputting data measured or analyzed by the control unit (21).

[0024] For example, the control unit (21) may include at least one processor and memory, and the memory may include a computer-readable storage medium or a computer-readable storage device. The memory may store instructions.

[0025] For example, the control unit (21) can perform functions and instructions for performing a multimodal data-based order reduction method of the turbine blade (9).

[0026] In one embodiment, the method of operation of an electronic device (2) for multimodal data-based order reduction may include a preprocessing operation (11) of receiving a discretized model and boundary conditions based on a geometric shape model for a designed turbine blade, an operation (12) of analyzing an overall order model based on information and parameters input from the preprocessing operation (11), an operation (13) of collecting the results of the overall order model analysis and building a database, an operation (14) of extracting a suitable orthogonal decomposition mode by performing singular value decomposition on the built database, an operation (15) of generating an order reduction model, an operation (16) of performing structural analysis of a turbine blade using the order reduction model, and a postprocessing operation (17) of processing, analyzing, and visualizing the analysis results to review various design proposals.

[0027] The preprocessing operation (11), which receives a discretized model and boundary conditions based on a geometric model of a designed turbine blade (9), may be an operation that performs element division necessary for analysis and sets parameters and boundary conditions.

[0028] Discretized elements include three-dimensional finite elements such as tetrahedrons, hexahedrons, and pyramid shapes, and discretized structures can be composed of models mixed with various elements as well as single elements. For example, as shown in FIG. 3, lower-order elements such as 4-node tetrahedrons and 8-node hexahedrons may be used, and higher-order elements such as 10-node tetrahedrons and 20-node hexahedrons may be used to discretize a turbine blade (9) containing multiple curved sections.

[0029] Boundary conditions can be input using predetermined parameters. In one embodiment, the parameters may consist of two or more types. For example, the parameters are load conditions (as illustrated in FIG. 4 or FIG. 5). ), rotation speed( It may include operating conditions of the turbine blade (9), such as fixed conditions. Meanwhile, the parameters may include material properties of the turbine blade (9), such as density and elastic modulus. The parameters may be set to values ​​within a design range.

[0030] The operation (12) of analyzing the entire order model based on information and parameters input from the preprocessing operation may be an operation of quantitatively analyzing the structural response of the turbine blade (9) based on set boundary conditions and material properties. In one embodiment, the operation (12) of analyzing the entire order model may include linear and nonlinear static analysis, time transient analysis, and forced vibration analysis. In such analysis processes, damping, rotational softening, and stress stiffness effects may be additionally considered.

[0031] In one embodiment, for a nonlinear static analysis, the control unit (21) uses a governing equation such as the following mathematical formula 1 to calculate the displacement increment ( ) can be calculated.

[0033]

[0034] ( : Material stiffness matrix, : Stress stiffness matrix, : Rotational softening matrix, : Displacement increment, : External force vector, : internal force vector, :: Newton-Raphson iterations)

[0036] In one embodiment, for linear time transient analysis, the control unit (21) uses a governing equation such as the following mathematical formula 2 to obtain a displacement vector ( ) can be calculated.

[0038]

[0039] ( : Mass damping matrix, : acceleration vector, : Attenuation matrix, : velocity vector, : Material stiffness matrix, : Rotational softening matrix, : Displacement vector, : External force vector)

[0041] In one embodiment, for nonlinear time transient analysis, a method similar to nonlinear static analysis may be applied to more accurately reflect the behavior of the structure. For example, the stiffness matrix may change in dependence on displacement, and iterative convergence calculations may be performed using the Newton-Raphson algorithm during the time-matching phase.

[0042] To accurately reflect nonlinear terms such as stiffness matrices and internal forces at each time step, an updated Lagrangian formula may be applied, similar to nonlinear static analysis. For example, in the case of nonlinear time transient analysis, calculations are performed by iteratively finding the solution to the time discretized equation, and the governing equation can be established as shown in Equation 3 below.

[0044]

[0045] ( : Mass damping matrix, : acceleration vector, : Attenuation matrix, : velocity vector, : Material stiffness matrix, : Stress stiffness matrix, : Rotational softening matrix, : Displacement vector, : External force vector, : Internal force vector)

[0047] In one embodiment, for forced vibration analysis, the displacement vector may be defined as in Equation 4 below, and the external force vector may be defined as in Equation 5 below. For forced vibration analysis, in order to derive steady-state vibration behavior and the resulting alternating stress, the response frequency and excitation frequency of the structure (as in Equations 4 and 5 below) It can be assumed that ) is the same.

[0049]

[0050] ( : Complex displacement amplitude, : Exceeding frequency)

[0052]

[0053] ( : Mass damping matrix, : Exceeding frequency)

[0055] By substituting the above mathematical equations 4 and 5 into the dynamic behavior equation of the structure, a governing equation such as the following mathematical equation 6 can be derived.

[0057]

[0058] ( : Mass damping matrix, : Attenuation matrix, : Material stiffness matrix, : Stress stiffness matrix, : Rotational softening matrix, : Complex displacement amplitude, : Mass damping matrix, : Exceeding frequency)

[0060] The operation (13) of collecting the results of the overall order model analysis and building a database may be an operation of storing and managing the results of static analysis, time transient analysis, and forced vibration analysis performed under various parameter conditions based on the above mathematical formula 6. The analysis results are stored in an electronic device (2) as shown in FIG. 6 and can be used as training data to create an order reduction model in the future.

[0061] The results of static analysis can be stored in a vector form with the total degree of freedom (DOF) of the structure, and the results of time transient analysis and forced vibration analysis can be stored in a matrix form with the number of times and frequencies as columns, respectively. These analysis results are built into an integrated database and can serve as a basis for selecting major modes during the future order reduction process.

[0062] The operation (14) of performing singular value decomposition (SVD) on the constructed database to extract a suitable orthogonal decomposition (POD) mode can be understood as an operation of selecting information of high importance among the various analysis information included in the database to construct an efficient order reduction model. The constructed database contains key displacement information that determines the structural response of the turbine blade (9), and based on this, an order reduction technique can be applied to increase analysis efficiency. As illustrated in FIG. 7, singular value decomposition can be applied to the constructed database. Singular value decomposition can be defined as shown in Equation 7 below.

[0064]

[0065] ( : Constructed database matrix, : Left singular matrix, : Singular value matrix, : Right singular matrix)

[0067] Left singular matrix ( ) includes the main behavioral components of the turbine blade (9), and the singular value matrix ( ) is a diagonal matrix representing the energy of each mode, which can be sorted in descending order. Singular value matrix( By analyzing ), the energy ratio for each mode can be verified, and a basis for the order reduction model can be formed by selecting a mode that conserves energy above a certain level. Accordingly, the left-singular matrix ( n modes can be selected starting from the first column of ), and a suitable orthogonal decomposition basis can be constructed as shown in Equation 8 below.

[0069]

[0071] The operation (15) of generating a reduced-order model may be an operation of converting the structural analysis equations into a reduced-order form using a suitable orthogonal decomposition basis. The suitable orthogonal decomposition basis derived from Equation 8 above may be a basis composed of some modes that can represent the overall behavior of the turbine blade (9). Based on this, the governing equations of the structural analysis can be converted using the Galerkin projection method to reduce the total degrees of freedom to n reduced degrees of freedom. Through this procedure, the computational efficiency of the structural analysis of the turbine blade can be increased. In one embodiment, for a nonlinear static analysis, the reduced-order governing equation may be expressed as shown in Equation 9 below.

[0073]

[0074] ( : Fit orthogonal resolution basis, : Material stiffness matrix, : Stress stiffness matrix, : Rotational softening matrix, : Reduced displacement, : External force vector, : internal force vector, :: Newton-Raphson iterations)

[0076] displacement increment ( ) is, as shown in Equation 10 below, a suitable orthogonal decomposition basis ( ) and reduction displacement( It can be defined as the product of ).

[0078]

[0079] ( : Fit orthogonal resolution basis, : Reduced displacement)

[0081] Similarly to the above, the governing equations of the time transient analysis and the forced vibration analysis can also have their degrees of freedom reduced by projecting them onto the coordinate system of the compatible orthogonal decomposition. For example, the time transient analysis projected onto the coordinate system of the compatible orthogonal decomposition is defined as shown in Equation 11 below, and the governing equation of the forced vibration analysis projected onto the coordinate system of the compatible orthogonal decomposition can be defined as shown in Equation 12 below.

[0083]

[0085]

[0087] The process from the preprocessing operation to the fitted orthogonal decomposition basis extraction operation is an offline phase that can be performed as a preliminary construction process for the order reduction model. In the offline phase, analysis data obtained under various parameter conditions can be collected, and an order reduction model can be trained and saved based on this data.

[0088] In the subsequent online stage, an operation (16) of performing a structural analysis of the turbine blade (9) using a reduced-order model can be performed. By utilizing a pre-built reduced-order model, the analysis for new parameter conditions can be performed quickly. In particular, since it deals with low-dimensional problems, various design proposals can be reviewed quickly during the design process of the turbine blade (9). Afterwards, a post-processing operation (17) of reviewing various design proposals by processing, analyzing, and visualizing the analysis results can be performed.

[0089] FIG. 8 is a diagram showing the deformation distribution of a full-order model and a reduced-order model according to one embodiment, and FIG. 9 is a graph showing the maximum displacement of the turbine blade tip over time in each of the full-order model and the reduced-order model according to one embodiment.

[0090] Referring to FIGS. 8 and 9, it can be seen that a reduced-order model (ROM) according to one embodiment can simulate a structural response similar to that of a full-order model (FOM). Even when reducing the order, the accuracy of the analysis is maintained, and it can be understood that the reduced-order model has high reliability compared to the analysis results of the full-order model. In one embodiment, the relative error between the result value using the reduced-order model and the result value using the full-order model can be defined as shown in Equation 13 below.

[0092]

[0093] ( : Relative error, : Displacement value calculated from the full-order model)

[0095] In mathematical equation 13, the displacement increment ( ) can be defined as shown in the following mathematical formula 14.

[0097]

[0098] ( : Displacement value calculated using the total order model, : Displacement value calculated using the order reduction model)

[0100] In one embodiment, the relative error derived through the mathematical formulas 13 and 14 ( ) is confirmed to be 0.034%, meaning that the known order reduction model provides almost the same level of interpretation accuracy as the full order model.

[0101] As shown in Fig. 8, the deformation distribution of the order reduction model and the full order model appears very similar, and it can be seen that the maximum deformation amount is also the same at the top of the turbine blade. It can be visually confirmed that a high degree of agreement with the original full order model can be maintained even when the order reduction technique is applied.

[0102] Referring to Fig. 9, it can be seen that the maximum displacement response at the turbine blade end between the reduced-order model and the full-order model is nearly identical, and that both models exhibit the same periodic behavior over time. It can be seen that the reduced-order model simulates the transient behavior of the turbine blade with the same accuracy as the full-order model.

[0103] In the present invention, computational efficiency was evaluated based on computation time and memory usage, and it was confirmed that the order reduction model reduced computation time by approximately 83.7% compared to the full order model. Specifically, the full order model took 35,469 seconds, while the order reduction model took 5,771 seconds. Regarding memory usage, the full order model used 5.7GB, whereas the order reduction model used only 1.2GB. Ultimately, it is confirmed that the order reduction method can reduce computational costs compared to the conventional case using the full order model.

[0104] Ultimately, by utilizing the present invention, the structural stability of turbine blades can be predicted quickly and accurately under various operating conditions, such as high pressure and high-speed rotation, thereby increasing the efficiency of analysis during the design phase. Furthermore, since it possesses versatility that allows for broad application to various analysis types, such as static analysis, time transient analysis, and forced vibration analysis, it can be applied not only to turbine blades but also to the analysis of various structures. Through this, the structural response of structures (e.g., turbine blades) can be systematically analyzed, contributing to design optimization and productivity improvement, and strengthening the reliability and durability of structures.

[0105] The embodiments described above may be implemented as hardware components, software components, and / or combinations of hardware and software components. For example, the devices, methods, and components described in the embodiments may be implemented using one or more general-purpose or special-purpose computers, such as, for example, a processor, a controller, an arithmetic logic unit (ALU), a digital signal processor, a microcomputer, a field programmable gate array (FPGA), a programmable logic unit (PLU), a microprocessor, or any other device capable of executing and responding to instructions. The processing unit may execute an operating system (OS) and one or more software applications executed on said operating system. Additionally, the processing unit may access, store, manipulate, process, and generate data in response to the execution of the software. For ease of understanding, the processing unit may be described as being used as a single unit, but those skilled in the art will understand that the processing unit may include multiple processing elements and / or multiple types of processing elements. For example, the processing unit may include multiple processors or one processor and one controller. Additionally, other processing configurations, such as parallel processors, are also possible.

[0106] Software may include computer programs, code, instructions, or a combination of one or more of these, and may configure a processing unit to operate as desired or command the processing unit independently or collectively. Software and / or data may be permanently or temporarily embodied in any type of machine, component, physical device, virtual equipment, computer storage medium or device, or transmitted signal wave so as to be interpreted by the processing unit or to provide instructions or data to the processing unit. Software may be distributed over networked computer systems and may be stored or executed in a distributed manner. Software and data may be stored on one or more computer-readable recording media.

[0107] The method according to the embodiment may be implemented in the form of program instructions that can be executed through various computer means and recorded on a computer-readable medium. The computer-readable medium may include program instructions, data files, data structures, etc., either alone or in combination. The program instructions recorded on the medium may be those specifically designed and configured for the embodiment, or they may be those known and available to those skilled in the art of computer software. Examples of computer-readable recording media include magnetic media such as hard disks, floppy disks, and magnetic tapes; optical recording media such as CD-ROMs and DVDs; magneto-optical media such as floptical disks; and hardware devices specifically configured to store and execute program instructions, such as ROM, RAM, and flash memory. Examples of program instructions include machine code, such as that generated by a compiler, as well as high-level language code that can be executed by a computer using an interpreter, etc. The hardware devices described above may be configured to operate as one or more software modules to perform the operation of the embodiment, and vice versa.

[0108] Although the embodiments have been described above with reference to the limited drawings, those skilled in the art can apply various technical modifications and variations based on the above. For example, suitable results may be achieved even if the described techniques are performed in a different order than described, and / or if the components of the described system, structure, device, circuit, etc. are combined or assembled in a form different from described, or replaced or substituted by other components or equivalents.

[0109] Therefore, other implementations, other embodiments, and equivalents to the claims also fall within the scope of the claims set forth below.

Claims

Claim 1 A multimodal data-based order reduction method for static and vibration structural analysis of a turbine blade, comprising: a preprocessing operation of receiving a discretized model and boundary conditions based on a geometric model of the turbine blade; an operation of analyzing an overall order model based on information and parameters input from the preprocessing operation; an operation of collecting the results of the overall order model analysis and constructing a database; an operation of extracting a suitable orthogonal decomposition mode by performing singular value decomposition on the constructed database; and an operation of generating an order reduction model. Claim 2 A method according to claim 1, wherein the operation of analyzing the entire order model is performed by including linear and nonlinear static analysis, time transient analysis, and forced vibration analysis. Claim 3 A method according to claim 1, wherein the parameter relates to the material properties or operating conditions of the turbine blade. Claim 4 In claim 3, the material properties of the turbine blade are the density or elastic modulus of the turbine blade. Claim 5 In claim 3, the operating condition of the turbine blade is the rotational speed of the turbine blade or the magnitude of the applied load. Claim 6 In claim 1, the operation of generating the order reduction model is performed using a Galerkin projection method. Claim 7 An electronic device for static and vibration structural analysis of a turbine blade, comprising: at least one processor; and a memory for storing instructions, wherein when the instructions are executed individually or collectively by the at least one processor, the electronic device is configured to receive a discretized model and boundary conditions based on a geometric model for a designed turbine blade, analyze a full-order model based on information and parameters input regarding the discretized model and boundary conditions, collect the results of the full-order model analysis to build a database, perform singular value decomposition on the built database to extract a suitable orthogonal decomposition mode, and generate a reduced-order model.

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