Shape Estimation for Additive Manufacturing Processes

Abstract

To receive a computer representation of a printable product part and a plan for the printable product part to be deposited using an additive manufacturing process.SOLUTION: The printable product part comprises an accumulation of material deposited by the additive manufacturing process. The plan comprises a tool-path representation of the printable product part and process parameters. Multiple as-printed shapes of the printable product part are determined after the printable product part has been deposited in accordance with the plan. Geometric differences between any of the multiple as-printed shapes and the computer representation of the product part are determined.SELECTED DRAWING: Figure 1

Description

[Technical field]

[0001] The present disclosure relates to the design and manufacture of mechanical parts. [Background technology]

[0002] Recent advances in additive manufacturing technology have spurred the development of powerful design methodologies, enabling designers to create highly complex functional parts. Summary of the Invention

[0003] The embodiments described herein involve receiving a computer representation of a printable product part and a plan for the printable product part to be deposited using an additive manufacturing process. The printable product part includes an accumulation of material deposited by the additive manufacturing process. The plan includes a toolpath representation of the printable product part and process parameters. A plurality of as-printed shapes of the printable product part are determined after the printable product part is deposited according to the plan. A contour difference between any of the plurality of as-printed shapes and the computer representation of the product part is determined.

[0004] The system involves a processor and a memory storing computer program instructions that, when executed by the processor, cause the processor to perform operations including receiving a computer representation of a printable product part and a plan for the printable product part to be deposited using an additive manufacturing process. The printable product part includes an accumulation of material deposited by the additive manufacturing process. The plan includes a toolpath representation of the printable product part and process parameters. A plurality of as-printed shapes of the printable product part are determined after the printable product part is deposited according to the plan. A contour difference between any of the plurality of as-printed shapes and the computer representation of the product part is determined.

[0005] An embodiment involves a non-transitory computer readable medium having computer program instructions stored thereon that, when executed by a processor, cause the processor to perform operations including receiving a computer representation of a printable product part and a plan for the printable product part to be deposited using an additive manufacturing process. The printable product part includes an accumulation of material deposited by the additive manufacturing process. The plan includes a toolpath representation of the printable product part and process parameters. A plurality of as-printed shapes of the printable product part are determined after the printable product part is deposited according to the plan. A contour difference between any of the plurality of as-printed shapes and the computer representation of the product part is determined.

[0006] The above summary is not intended to describe each embodiment or every implementation. A more complete understanding will become apparent and understood by reference to the following detailed description and claims, taken in conjunction with the accompanying drawings. [Brief description of the drawings]

[0007] [Figure 1] 1 illustrates a process for predicting the shape of an as-printed part according to embodiments described herein. [Figure 2A] 1 illustrates an exemplary drop-on-demand magnetohydrodynamic (MHD) deposition system according to embodiments described herein. [Figure 2B] 1 illustrates a simulation model showing the magnetic field generated by a pulsed magnetic coil as well as the volume fraction of ejected liquid aluminum according to embodiments described herein. [Diagram 3] 1 shows a spherical shaped hot liquid droplet falling onto a substrate of any shape according to embodiments described herein. [Figure 4A] 13 shows simulation and experimental results at different times during the solidification process according to embodiments described herein. [Figure 4B]13 shows simulation and experimental results at different times during the solidification process according to embodiments described herein. [Figure 4C] 13 shows simulation and experimental results at different times during the solidification process according to embodiments described herein. [Figure 4D] 13 shows simulation and experimental results at different times during the solidification process according to embodiments described herein. [Figure 5A] 1 illustrates an example of a training set according to embodiments described herein. [Figure 5B] 1 illustrates an example of a training set according to embodiments described herein. [Figure 6A] 1 shows an example of a network for predicting as-printed shapes according to embodiments described herein. [Figure 6B] 1 illustrates a fully connected feed-forward neural network used to predict the state of the solidified droplet and substrate according to embodiments described herein. [Figure 7] 1 shows a plot of an activation function according to an embodiment described herein. [Figure 8] 13 shows a plot of a counting function for counting non-zero voxels according to embodiments described herein. [Figure 9A] 1 illustrates an example neural network input according to embodiments described herein. [Figure 9B] 1 illustrates a neural network prediction based on input, according to embodiments described herein. [Figure 9C] 1 shows the correct solution from a multiphysics solver according to an embodiment described herein. [Figure 10A] 1 shows a schematic plot of a workflow according to an embodiment described herein. [Figure 10B] 1 shows a schematic plot of a workflow according to an embodiment described herein. [Figure 10C]1 shows a schematic plot of a workflow according to an embodiment described herein. [Figure 10D] 1 shows a schematic plot of a workflow according to an embodiment described herein. [Figure 10E] 1 shows a schematic plot of a workflow according to an embodiment described herein. [Figure 11] 13 shows a volume plot of a calculated kernel according to an embodiment described herein. [Figure 12] 1 shows an example of a network for predicting as-printed shapes according to embodiments described herein. [Figure 13A] 1 illustrates exemplary inputs and outputs of two different shapes according to embodiments described herein. [Figure 13B] 1 illustrates exemplary inputs and outputs of two different shapes according to embodiments described herein. [Figure 14A] 1 illustrates exemplary inputs and outputs of two different shapes according to embodiments described herein. [Figure 14B] 1 illustrates exemplary inputs and outputs of two different shapes according to embodiments described herein. [Figure 15] FIG. 1 illustrates a block diagram of a system in which the embodiments described herein may be implemented. The drawings are not necessarily to scale. Like numbers used in the drawings refer to like components. However, it will be understood that the use of a number to refer to a component in a given figure is not intended to limit the component in another figure labeled with the same number. DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS

[0008] This disclosure relates to additive manufacturing (AM). Many such methods often operate under the assumption that any designed shape can be machined and that the resulting part will match the designed shape within an acceptable tolerance. However, in reality, the selected AM process, the mechanical properties of the printer, or the materials used may pose significant limitations to the printability of a particular design within an acceptable tolerance. Effective geometric modeling of the as-printed part shape would aid process planning by eliminating costly trial and error with multiple builds and rapidly converging on parameter values ​​that result in acceptable part quality.

[0009] Uncertainties in any manufacturing process result in deviations between the nominal design and the workpiece. Nominal design shapes and material layouts are constantly altered in the AM process, and shape variations can result in undesirable effects such as (unintended) porosity and surface roughness. These in turn can adversely affect residual stresses, fatigue failure mechanisms such as crack initiation, and / or bulk mechanical properties, leading to long-term performance degradation. Metal parts designed for high-stress applications can be fully densified with smooth surfaces to minimize the chance of breakage during use. While such properties can also be achieved with machining, the complexity of geometries that are achievable using AM allows for the production of functional, high-performance lightweight parts that may otherwise be difficult to fabricate. This feature of metal AM creates the impetus for the need to understand the relationship between the nominal design and its corresponding various classes of shapes that arise due to a combination of selected AM process parameters and manufacturing errors. It is to be understood that while the embodiments described herein focus on selected metal AM processes described herein, they can be used in conjunction with AM processes using any material. For example, the processes and systems described herein can be used with polymer AM.

[0010] For metal AM, microstructural imaging by Light Optical Microscopy (LOM) and Scanning Electron Microscopy (SEM) is used to study microstructural details such as grain morphology, grain size, and phase identification for powder bed fusion and directed energy deposition processes. The columnar grain orientation seen in these processes is significantly influenced by the combination of scanning strategy and applied energy to induce material phase changes, which are key for AM processes. Experimental analysis to map process parameters to specific manufacturing-driven structures and material variations is performed for each material and process combination in the metal AM process.

[0011] It should be understood that AM does not have to be a stand-alone process, but may be followed by heat treatment to relieve residual stresses and / or improve mechanical performance, and machining to improve part surface quality and / or to remove support material. Applications involving post-processing may also require planning for post-processing operations, such as support material removal using cutting tools and / or finishing of rough part surfaces. Equipment manufacturers are interested in geometry characteristics such as minimum feature size / resolution, surface roughness, and / or accuracy to ensure that the overall machined geometry (excluding support material) is as close as possible to the nominal design. It may therefore be useful to estimate the machined geometry corresponding to the nominal design as a function of AM process parameters so that critical characteristics such as porosity, roughness, and / or geometry deviations from the nominal design can be characterized prior to machining. To date, little attention has been paid to computational modeling and representation of the as-manufactured part geometry. Effective geometry modeling of the as-manufactured geometry would aid process planning by eliminating costly trial and error with multiple builds.

[0012] The embodiments described herein involve using one or more models to estimate the processed geometry, also referred to as the as-printed part, at different length scales. The shape and structure of the accumulated material at the length scale of the material deposition (typically tens to hundreds of micrometers) contribute to the overall shape of the as-printed part. However, it may not be necessary to model the detailed shape at the length scale of the material accumulation to represent the as-printed part. In some configurations, the model uses a neural network to predict the state of the accumulated material and the substrate after adding material onto the substrate. After deposition, the state of the deposited and accumulated material is added to the state of the substrate. At the next deposition event, the total state of the previously accumulated material and the substrate is considered the state of the substrate. In other words, the deposition changes the state of the substrate. The state of the part includes multiple states of the deposited material along the toolpath. Variables describing the physical state of the accumulated material and the substrate may include, but are not limited to, spatial distribution of solid and liquid phases, spatial distribution of temperature, pressure, and flow rate, etc. The shape of the part includes the spatial distribution of the solid phase of the part.

[0013] According to various embodiments, the first model is a high-fidelity physical model of a single deposition event. To model the built part, a solver may need to be invoked at every point along the toolpath. The solver calculates the state of the deposited and accumulated material and / or substrate after the deposition event. The high-fidelity solver receives as input the initial state of the smallest printable features and the substrate integrated in time through the deposition, outputting the final state of the smallest printable features and substrate. The high-fidelity solver is a very accurate solution, but the accuracy comes at a high computational cost. For example, a simulation of deposition at a single location (using such a model) can take more than an hour on a high-end modern desktop computer. In the case of droplet-based additive manufacturing, the deposited material at a location corresponds to a metal droplet that solidifies upon contact with the substrate.

[0014] Using a multiphysics solver involves satisfying certain numerical constraints during the integration process to solve the (partial) differential equations that are the laws of physics of material accumulation. The constraints come from the mathematical formulation of the multiphysics solver. If these constraints are not satisfied, the solver may incur instabilities, resulting in divisions by zero or integration steps that are effectively zero. In this situation, the solver cannot converge to a solution. However, neural networks as surrogate models can be orders of magnitude faster than multiphysics models, since they do not involve time integration. Surrogate models also cannot be limited by constraints, unless they are explicitly coded. Surrogate models input the distribution of the liquid and solid phases and the initial state of the substrate, and directly output the distribution after deposition and the state of the substrate. The neural network of the surrogate model is trained using one or more training sets. Once the network is trained (which is a one-time task), the predictions summarize a set of basic linear algebraic operations that can be implemented very efficiently through parallelization (on CPU and / or GPU cores). Thus, the trained network can perform calculations very quickly. According to various embodiments described herein, the training data set can be obtained through a combination of high-fidelity models (e.g., thermal-fluid multiphase flow simulations), simplified models (e.g., steady-state shapes of droplets), and / or actual experiments. If the training data is obtained through a computational model, the model may involve calibration with experiments prior to data collection. Calibration can be performed by matching the output of the model with the experiment.

[0015] Although droplet-based additive manufacturing is described herein, it should be understood that the techniques described may be applied to any type of additive manufacturing process. The output and substrate state are described by the spatial distribution of solid, liquid, and temperature. A trained neural network receives the spatial distribution of solid, liquid, and temperature before deposition and the output of the spatial distribution of solid, liquid, and temperature after deposition. The run time of this network on a desktop is about 0.005 seconds. The error in the accuracy of the prediction is less than 10 percent compared to the high-fidelity solution. The run time of the high-fidelity multiphysics solver for the same deposition is 20 minutes. Given the toolpath for machining the part and the parameters for estimating the local material accumulation using the neural network, the neural network is repeatedly called at positions along the toolpath to estimate the shape of the accumulated material throughout the manufacturing plan. This approach explicitly models the machining shape at multiple length scales (micrometers to meters). The resulting shape is a high-resolution representation of the as-printed part.

[0016] The second model uses a convolutional neural network to directly estimate the as-printed shape at the part length scale without explicitly modeling the material accumulation at the deposition length scale. The second model inputs a toolpath representation of the part and outputs a spatial distribution of the probability of local material accumulation. The printing process is always uncertain. Two as-printed shapes of the same part will never be the same event even with the same toolpath and process parameters. The uncertainty comes from the uncertainty exhibited in the process. This can be mechanical or electrical noise, random disturbances, material variations, etc. Uncertainty in the process is translated into uncertainty in the as-printed part shape. The convolutional network captures this uncertainty in a computationally efficient manner. An alternative efficient implementation of this operation can use a GPU accelerated algorithm to calculate the convolutions using the Fast Fourier Transform.

[0017] FIG. 1 illustrates a process for predicting an as-printed part shape according to embodiments described herein. A computer representation of a printable product part and a plan for the printable product part to be deposited using an additive manufacturing process are received (110). According to various embodiments, the printable product part includes an accumulation of material deposited by the additive manufacturing process. While the embodiments described herein are applicable to any type of additive manufacturing process, in some cases, the additive manufacturing process includes a plurality of liquid metal droplets that fuse and solidify according to the plan. The plan may include a toolpath representation of the printable product part.

[0018] After being deposited according to the printable product part plan, a plurality of as-printed shapes of the printable product part are determined (120). According to various embodiments, the plurality of as-printed shapes are determined based on at least one physical state of one or both of the at least one deposition material and the substrate. The physical state may include, for example, one or more of a spatial distribution of solid and liquid phases of the material, a spatial distribution of temperature, pressure, and flow rate. According to various embodiments, each of the plurality of as-printed shapes includes a spatial distribution of a solid phase of the as-printed shape.

[0019] A contour difference between any of the multiple as-printed shapes and the computer representation of the product part is determined 130. In some cases, a Boolean difference between the computer representation of the product part and some or all of the as-printed shape of the product part is calculated. The contour difference may be displayed on a user interface.

[0020] In some cases, the manufacturing error from a plurality of as-printed shapes of the product part is determined based on a statistical analysis. According to various embodiments, the training data is generated using one or more of synthetically generated data and experimental data. Based on the training data, a model can be constructed for determining the manufacturing uncertainty of the product part. The manufacturing uncertainty of the product part is determined based on the model.

[0021] According to various embodiments described herein, receiving a plan for a printable product part includes receiving at least one model that determines material accumulation in successive steps of the plan, where the at least one model may include one or more of a multiphysics model, a proxy model for the multiphysics model, a machine learning model as a proxy for the multiphysics model, and a reduced order model as a proxy for the multiphysics model.

[0022] The embodiments described herein use the classical idea of ​​predictive estimation with uncertainty quantification. Uncertainties in manufacturing processes are quantified using simulated data obtained by modeling the physics of material deposition and accumulation. The method also works with data obtained by scanning the processed parts. First, a simulated data set of simple shape is generated for a fixed set of process parameters, where N is the number of data points in the set and each data point D={{T l ,P l},l=1,...N} are a pair of 3D arrays corresponding to the mapping of toolpaths and their probabilities to a spatial distribution. The toolpath T is the sampled representation function, and the array P is the spatial distribution of the local material accumulation probability (all

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[0023] The embodiments described herein can use a drop-on-demand magnetohydrodynamic (MHD) deposition system as shown in FIG. 2A. It is understood that other types of printing processes and / or systems may be used. In the MHD method, a spooled solid metal wire (e.g., aluminum wire) 210 and gas 230 are continuously fed into a ceramic heating chamber 220 of a magnetohydrodynamic print head, where they are resistively melted to form a reservoir of liquid metal that feeds into the ejection chamber via capillary forces. A coil 225 at least partially surrounds the ejection chamber and is electrically pulsed (240), creating a transient magnetic field that permeates the liquid metal and induces a closed-loop transient electric field therein. The electric field creates a circulating current density that back-couples to the transient magnetic field, creating a magnetohydrodynamic Lorentz force density in the chamber. The radial component of this force creates a pressure that acts to eject liquid metal droplets from the nozzle opening. The ejected droplets travel to a substrate 250 where they fuse, solidify, and form an elongated solid structure. The three-dimensional structures are printed layer by layer using a moving substrate 250 controlled by a controller that allows for precise pattern deposition. The process is controlled by a controller 260. Figure 2B shows a simulation model illustrating the magnetic field generated by the pulsed magnetic coil as well as the volume fraction of the ejected liquid aluminum.

[0024] For high fidelity models, the process of droplet-based AM can be modeled as a sequence of localized single-droplet events. A hot droplet of spherical shape 340 falling onto an arbitrarily shaped substrate 320 is shown in FIG. 3. According to the embodiments described herein, the substrate surface temperature is spatially uniform and the substrate is substantially solid on average. At least a portion of the substrate may be in a liquid state. A large portion of the liquid in the substrate may violate the localization assumption. Therefore, it can be assumed that the amount of liquid phase in the substrate is much smaller than the amount of solid phase. After hitting the substrate, the droplet bottom merges with the substrate surface. These assumptions can be made over a controlled range of process parameters. The droplet solidification front starts to propagate from the bottom of the droplet, where the temperature is relatively cool. During solidification, the top of the droplet may bounce several times before finally forming a steady state. The droplet may also remelt the underlying substrate solid phase. The remelting can be assumed to be a highly localized event. Residual stresses can cause warping in the final profile. It can be assumed that the temperature difference between the droplet and the substrate is not significant enough to produce an explainable warping. Eliminating residual stresses allows droplet-based AM to be modeled using a coupled system of viscous fluid and heat transfer equations, where solidification is described by a momentum loss term in the momentum conservation equation. The momentum loss term effectively freezes the liquid when the temperature falls below the melting point.

[0025] Consider the system of incompressible Navier-Stokes equations for multiphase flow as shown in (2).

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[0026] All physical parameters of the model can be related to the physical parameters of a given alloy. These can be measured, for example, from direct experiment. Nevertheless, it may be possible to distinguish between the two parameters only by indirect comparison of the simulation results with the experiment. A high-speed camera may be used to measure the time sequence of droplet solidification on a flat surface. By matching the droplet shape at any given time step, the numerical coefficients in the Darcy terms, C and C, can be determined. Lee The constants can be identified. Also, the multiphase flow equation system can be solved by using the finite volume method for spatial discretization and the volume of fluid (VOF) method for interface tracking. The spatial resolution of the computational domain is about 16 μm or better. The initial condition of the simulation is a droplet on a surface. The initial droplet consists of a pure liquid phase and an initial substrate with a pure solid phase. Numerical simulations of a single droplet solidifying on a flat substrate are shown in Figures 4A-4D, which show a comparison of the simulation results with experiments for different times in the solidification process.

[0027] FIG. 4A shows the simulation result 410 and the experimental result 420 at t=0 ms. As can be seen, the droplet 412 is in contact with the substrate 414. In the experimental result 420, the results below the line 425 are ignored because they are reflections. FIG. 4B shows the simulation result 430 and the experimental result 440 at t=0.3 ms. As can be seen, the droplet 432 is in contact with the substrate 434 and starts to solidify at the interface between the droplet 432 and the substrate 434. In the experimental result 440, the results below the line 445 are ignored because they are reflections. FIG. 4C shows the simulation result 450 and the experimental result 460 at t=0.9 ms. Compared to the previous results, more droplets 452 have solidified at the interface between the droplet 452 and the substrate 454. In the experimental result 460, the results below the line 465 are ignored because they are reflections. FIG. 4D shows the simulation result 470 and the experimental result 480 at t=2.7 ms. At the timestamp shown in Figure 4D, most of the droplet 472 has solidified on the substrate 474. In the experimental results 480, results below line 485 are ignored as they are reflections. The close agreement of the simulated and observed droplet fusion indicates that the multiphysics model accurately captures the shape of the solidifying droplets and the time scale of solidification.

[0028] As described herein, predicting the as-printed part shape using a multiphysics model involves the solution of a highly nonlinear, time-dependent system of partial differential equations that exploit complex physics. For example, droplet-based additive manufacturing involves the numerical solution of multiphase flow and heat transfer in three dimensions. Solving such a complex system of equations on a top-of-the-line modern desktop computer is computationally expensive. For example, simulating the deposition of a single droplet using a high-fidelity model can involve approximately one hour of computer time. Millions (or even billions) of droplets may be used to simulate part construction. Thus, to address such complexity, the embodiments described herein involve an efficient reduced-order surrogate model for estimating the as-printed shape using machine learning. This reduced-order model has the advantage of reducing the computational time to simulate droplet deposition. Using this highly efficient reduced-order model, the shape of the printed part can be estimated in minutes. According to various embodiments described herein, the two models are combined into one model, where the as-printed part geometry is first estimated using a surrogate model, and certain portions of the contour are solved using the multiphysics model. Furthermore, using a multiphysics solver may involve satisfying certain numerical constraints during the integration process. The constraints come from the mathematical formulation of the multiphysics solver. If these constraints are not met, the solver may suffer from instabilities, resulting in divisions by zero or integration steps that are essentially zero. In this situation, the solver will not converge to a solution.

[0029] For a surrogate model given a sufficiently large and representative training set, the deep neural network can interpolate the multi-dimensional data. The deep neural network may be trained to predict the shape of the deposited droplet after, for example, every Δt=0.01 s. The time step Δt may correspond to a deposition frequency of 100 Hz. First, a training set is generated by depositing droplets on various surfaces. Examples of training sets are shown in Figs. 5A and 5B. Fig. 5A shows the input at t=0 s. A droplet 510 is deposited on a substrate 530. A solid-phase droplet 520 has already been deposited on the substrate 530 and solidified. Fig. 5B shows the distribution of the solid and liquid phases at t=0.01 s. As can be observed, the deposited droplet 510 from Fig. 5A is now within the solid phase 540 on the substrate 550.

[0030] As described herein, the multiphysics solver outputs simulated spatial distributions of solid, liquid, and gas phases, as well as spatial distributions of temperature, pressure, and / or flow velocity. To reduce the input size to the neural network, the spatial distributions of the solid and liquid phases are expressed as a function of the field α=T(α solid -α liquid ) and combined with temperature. The neural network is i solid , α i liquid}, i=1, ..., N set}, where the superscripts correspond to pair indices and N set is the total number of pairs in the set.

[0031] Figure 6A shows an example of a network predicting the as-printed shape. The training set for the second neural network can be obtained experimentally or numerically. Experimental data collection involves printing many parts of the same geometry. The printed parts then need to be scanned and digitized. The probability of local material accumulation is estimated by counting, for every voxel, the number of times it is filled with material. If the training data is collected from a numerical simulation, only the last step is needed, since the numerical model directly outputs a digital representation of the shape. Numerical models do not have uncertainties unless explicitly specified. To model the uncertainties, random numbers need to be added to the parameters of the model for each deposition event. The first neural network can be used to accelerate the numerical data collection of the second model.

[0032] FIG. 6B shows a fully connected feed-forward neural network used to predict the state of the solidified droplet and the substrate. The domain dimensions are reduced inside the network (to reduce the amount of GPU memory required). As shown in FIG. 6B, the neural network is composed of a series of matrix multiplications followed by a nonlinear transformation. In machine learning nomenclature, the matrix multiplications are called linear layers and the nonlinear parts are called activation functions. A neural network should have at least one linear layer followed by an activation function. Training a neural network may involve inputting a sufficient training set. The size of the training set can be reduced by encoding physical constraints in the loss function and activation function. The loss function is minimized during the neural network training process. Physical constraints can also be encoded inside the activation function.

[0033] The custom activation function is T min <|α i、j、k | <T max is used to ensure that i、j、k are the inputs and outputs of the layer. The activation function is defined as (10).

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[0034] The network is trained by minimizing the sum of two cost functions. The first cost function is the difference between the output of the network and a target value. The second term is the difference between the masses of the input and output. The second constraint enforces the conservation of the total mass, since the total mass of the solid and liquid phases is conserved between the input and output. The total mass is obtained using the function shown in (13) below.

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[0035] 9A-9C show an example input (FIG. 9A), the neural network's predictions (FIG. 9B), and the correct answer from the multiphysics solver (FIG. 9C). In this example, the neural network set= 10000 samples. Shown are solid phases 930, 932, 934 and liquid phases 910, 920, 922, 924. It can be observed that the surrogate model (FIG. 9B) produces very similar output to the multiphysics solver (FIG. 9C) in about 0.1 seconds, whereas generating an individual droplet fusion simulation using the multiphysics solver takes about an hour.

[0036] According to the embodiments described herein, the manufacturing uncertainty of a product part may be determined to predict the as-printed shape. The kernel may be found in equation (1) by training a Convolutional Neural Network (CNN) with a single convolutional layer and zero bias. Other methods may include, for example, computing the convolution using a Fast Fourier Transform on a graphics processing unit. First, a surrogate model is used to generate a training set. The training set is generated by approximating the toolpath with a spatial distribution probability D={{Tl,P l}, l=1,...,N}. ijk ,i=1,...,N j ,k=1,...,N k} is an array of binary numbers

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[0037] According to the embodiments described herein, the CNN is trained with a set of 100 pairs of toolpaths and corresponding probability fields. The weights of the convolution layers are extracted after training, which provides the solution of the deconvolution problem to estimate the kernel K. A volume plot of the computed kernel is shown in FIG. 11. Using this kernel, the as-manufactured probability field can be determined by using (1). Then, taking the appropriate sublevel set of the field (P) for the selected probability (p) will give an estimate of the as-manufactured shape (for that probability). FIG. 12 shows an example of a network predicting the as-printed shape according to the embodiments described herein. FIG. 13A shows an example input and FIG. 13B shows an example output for a first shape. Similarly, FIG. 14A shows an example input and FIG. 14B shows an example output for a second shape.

[0038] The above method can be implemented in a computer using well-known computer processors, memory units, storage devices, computer software and other components. A high-level block diagram of such a computer is shown in FIG. 15. The computer 1500 includes a processor 1510 that controls the overall operation of the computer 1500 by executing computer program instructions that define such operation. It should be understood that the processor 1510 may include any type of device capable of executing instructions. For example, the processor 1510 may include one or more of a central processing unit (CPU), a graphical processing unit (GPU), a field-programmable gate array (FPGA) and an application-specific integrated circuit (ASIC). The computer program instructions may be stored in a storage device 1520 (e.g., a magnetic disk) and loaded into the memory 1530 when execution of the computer program instructions is desired. Thus, the steps of the methods described herein may be defined by computer program instructions stored in the memory 1530 and controlled by the processor 1510 that executes the computer program instructions. The computer 1500 may include one or more network interfaces 1550 for communicating with other devices over a network. The computer 1500 also includes a user interface 1560 that allows user interaction with the computer 1500. The user interface 1560 may include I / O devices 1562 (e.g., a keyboard, a mouse, speakers, buttons, etc.) to allow a user to interact with the computer. Such input / output devices 1562 may be used in conjunction with a set of computer programs according to embodiments described herein. The user interface also includes a display 1564.According to various embodiments, FIG. 15 is a high-level representation of possible components of a computer for illustrative purposes, and a computer may include other components.

[0039] Unless otherwise indicated, all numbers expressing shapes, sizes, quantities, and physical properties used in the specification and claims are to be understood in all instances as modified by the term "about." Thus, unless otherwise indicated, the numerical parameters set forth in the foregoing specification and appended claims are approximations that may vary depending on the desired properties one of ordinary skill in the art would obtain using the teachings disclosed herein. The use of numerical ranges by endpoints includes all numbers within that range (e.g., 1 to 5 includes 1, 1.5, 2, 2.75, 3, 3.80, 4, and 5) and any range within that range.

[0040] The various embodiments described above may be implemented using circuits and / or software modules that interact to provide particular results. Those skilled in the computing arts can readily implement such described functionality, either at a modular level or as a whole, using knowledge commonly known in the art. For example, the flow charts shown herein may be used to create computer readable instructions / code for execution by a processor. Such instructions may be stored on a computer readable medium and transferred to a processor for execution as known in the art. The structures and procedures shown above are merely representative examples of implementations that may be used to facilitate such embodiments.

[0041] The foregoing description of exemplary embodiments has been presented for purposes of illustration and description. It is not intended to be exhaustive or to limit the inventive concepts to the precise form disclosed. Many modifications and variations are possible in light of the above teachings. Any or all features of the disclosed embodiments may be applied individually or in any combination and are not intended to be limiting but are purely exemplary. It is intended that the scope of the invention be limited not by the detailed description but by the claims appended hereto.

Claims

1. 1. A method comprising: receiving a computer representation of a printable product part and a plan for the printable product part to be deposited using an additive manufacturing process, the printable product part including an accumulation of material deposited by the additive manufacturing process, the plan including a representation of process parameters and tool paths used in building the printable product part; determining a plurality of as-printed shapes of the printable product part after the printable product part is deposited according to the plan, wherein determining the plurality of as-printed shapes comprises performing a physics-informed simulation in which uncertainty is introduced into the tool path, and for each of the plurality of as-printed shapes, the physics-informed simulation comprises: for each deposition event of the plan, inputting initial conditions of distribution of liquid and solid phases into a feedforward neural network having a loss function and an activation function based on physically-informed information, the feedforward neural network outputting a spatial distribution of substrate after the deposition event; Obtaining a spatial distribution of a solid phase of the as-printed shape estimated by performing a convolution of a kernel with the tool path; and and determining a contour difference between the spatial distribution of the solid phase of any of the plurality of as-printed shapes and the computer representation of the product part; and using the contour difference in the additive manufacturing process to build the printable product part with acceptable part quality.

2. The method of claim 1 , wherein the additive manufacturing process comprises a plurality of liquid metal droplets that fuse and solidify according to the plan.

3. The method of claim 1 , further comprising determining a manufacturing tolerance from the plurality of as-printed features of the product part based on a statistical analysis.

4. 10. The method of claim 1, further comprising determining the plurality of as-printed shapes based on at least one physical condition of one or both of at least one of the deposited material and the substrate.

5. The method of claim 4 , wherein the at least one physical state comprises one or more of a spatial distribution of solid and liquid phases of the material, a spatial distribution of temperature, pressure, and flow rate.

6. 10. The method of claim 1, comprising generating training data for the feedforward neural network using one or more of synthetically generated data and experimental data.

7. The method of claim 1 , comprising calculating the difference between the computer representation of the product part and some or all of the as-printed shape of the product part.

8. The method of claim 7 further comprising displaying the contour differences on a user interface.

9. The method of claim 1 , wherein the kernels are estimated by training a convolutional neural network with a single convolutional layer.

10. 1. A system comprising: A processor; and a memory storing computer program instructions which, when executed by the processor, cause the processor to: receiving a computer representation of a printable product part and a plan for the printable product part to be deposited using an additive manufacturing process, the printable product part including an accumulation of material deposited by the additive manufacturing process, the plan including a representation of process parameters and tool paths used in building the printable product part; determining a plurality of as-printed shapes of the printable product part after the printable product part is deposited according to the plan, wherein determining the plurality of as-printed shapes comprises performing a physics-informed simulation in which uncertainty is introduced into the tool path, and for each of the plurality of as-printed shapes, the physics-informed simulation comprises: for each deposition event of the plan, inputting initial conditions of distribution of liquid and solid phases into a feedforward neural network having a loss function and an activation function based on physically-informed information, the feedforward neural network outputting a spatial distribution of substrate after the deposition event; Obtaining a spatial distribution of a solid phase of the as-printed shape estimated by performing a convolution of a kernel with the tool path; and and determining a contour difference between the spatial distribution of the solid phase of any of the plurality of as-printed shapes and the computer representation of the product part, wherein the contour difference is used in the additive manufacturing process to build the printable product part with acceptable part quality.

11. 11. The system of claim 10, wherein receiving the plan for the printable product part includes receiving at least one of a multiphysics model and a machine learning model as a surrogate for the multiphysics model.

12. The system of claim 10 , further comprising a user interface configured to display differences in appearance.

13. The system of claim 10 , wherein the additive manufacturing process comprises a plurality of liquid metal droplets that fuse and solidify according to the plan.

14. The system of claim 10 , wherein the processor is further configured to determine a manufacturing error from the plurality of as-printed shapes of the product part based on a statistical analysis.

15. 11. The system of claim 10, wherein the processor is further configured to determine the plurality of as-printed shapes based on at least one physical state of one or both of at least one of the deposited material and the substrate.

16. The system of claim 15 , wherein the at least one physical state comprises one or more of a spatial distribution of solid and liquid phases of the material, a spatial distribution of temperature, pressure, and flow rate.

17. 16. The system of claim 15, wherein each of the plurality of as-printed features comprises a spatial distribution of a solid phase of the as-printed feature.

18. A non-transitory computer readable medium storing computer program instructions that, when executed by a processor, cause the processor to: receiving a computer representation of a printable product part and a plan for the printable product part to be deposited using an additive manufacturing process, the printable product part including an accumulation of material deposited by the additive manufacturing process, the plan including a representation of process parameters and tool paths used in building the printable product part; determining a plurality of as-printed shapes of the printable product part after the printable product part is deposited according to the plan, wherein determining the plurality of as-printed shapes comprises performing a physics-informed simulation in which uncertainty is introduced into the tool path, and for each of the plurality of as-printed shapes, the physics-informed simulation comprises: For each deposition event in the plan, inputting initial conditions of distribution of liquid and solid phases into a feedforward neural network having a loss function and an activation function based on physically-informed information, the feedforward neural network outputting a spatial distribution of substrate after the deposition event; Obtaining a spatial distribution of a solid phase of the as-printed shape estimated by performing a convolution of a kernel with the tool path; and and determining a contour difference between the spatial distribution of the solid phase of any of the plurality of as-printed shapes and the computer representation of the product part; and using the contour differences in the additive manufacturing process to build the printable product part with acceptable part quality.

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