Pub Date : 2025-01-27DOI: 10.1016/j.cma.2025.117759
Wietse M. Boon , Nicola R. Franco , Alessio Fumagalli
We consider a mixed formulation of parametrized elasticity problems in terms of stress, displacement, and rotation. The latter two variables act as Lagrange multipliers to enforce the conservation of linear and angular momentum. The resulting system is computationally demanding to solve directly, especially if various instances of the model parameters need to be investigated. We therefore propose a reduced order modeling strategy that efficiently produces an approximate solution, while guaranteeing conservation of linear and angular momentum in the computed stress. First, we obtain a stress field that balances the body and the boundary forces by solving a triangular system, generated with the use of a spanning tree in the grid. Second, a trained neural network is employed to rapidly compute a correction without affecting the conservation equations. The displacement and rotation fields can be obtained by post-processing. The potential of the approach is highlighted by three numerical test cases, including a three-dimensional and a non-linear model.
{"title":"Neural network solvers for parametrized elasticity problems that conserve linear and angular momentum","authors":"Wietse M. Boon , Nicola R. Franco , Alessio Fumagalli","doi":"10.1016/j.cma.2025.117759","DOIUrl":"10.1016/j.cma.2025.117759","url":null,"abstract":"<div><div>We consider a mixed formulation of parametrized elasticity problems in terms of stress, displacement, and rotation. The latter two variables act as Lagrange multipliers to enforce the conservation of linear and angular momentum. The resulting system is computationally demanding to solve directly, especially if various instances of the model parameters need to be investigated. We therefore propose a reduced order modeling strategy that efficiently produces an approximate solution, while guaranteeing conservation of linear and angular momentum in the computed stress. First, we obtain a stress field that balances the body and the boundary forces by solving a triangular system, generated with the use of a spanning tree in the grid. Second, a trained neural network is employed to rapidly compute a correction without affecting the conservation equations. The displacement and rotation fields can be obtained by post-processing. The potential of the approach is highlighted by three numerical test cases, including a three-dimensional and a non-linear model.</div></div>","PeriodicalId":55222,"journal":{"name":"Computer Methods in Applied Mechanics and Engineering","volume":"437 ","pages":"Article 117759"},"PeriodicalIF":6.9,"publicationDate":"2025-01-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143071662","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-01-26DOI: 10.1016/j.cma.2025.117751
Jizu Huang, Rukang You, Tao Zhou
Multi-scale deep neural networks (MscaleDNNs) with downing-scaling mapping have demonstrated superiority over traditional DNNs in approximating target functions characterized by high frequency features. However, the performance of MscaleDNNs heavily depends on the parameters in the downing-scaling mapping, which limits their broader application. In this work, we establish a fitting error bound to explain why MscaleDNNs are advantageous for approximating high frequency functions. Building on this insight, we construct a hybrid feature embedding to enhance the accuracy and robustness of the downing-scaling mapping. To reduce the dependency of MscaleDNNs on parameters in the downing-scaling mapping, we propose frequency-adaptive MscaleDNNs, which adaptively adjust these parameters based on a posterior error estimate that captures the frequency information of the fitted functions. Numerical examples, including wave propagation and the propagation of a localized solution of the Schrödinger equation with a smooth potential near the semi-classical limit, are presented. These examples demonstrate that the frequency-adaptive MscaleDNNs improve accuracy by two to three orders of magnitude compared to standard MscaleDNNs.
{"title":"Frequency-adaptive multi-scale deep neural networks","authors":"Jizu Huang, Rukang You, Tao Zhou","doi":"10.1016/j.cma.2025.117751","DOIUrl":"10.1016/j.cma.2025.117751","url":null,"abstract":"<div><div>Multi-scale deep neural networks (MscaleDNNs) with downing-scaling mapping have demonstrated superiority over traditional DNNs in approximating target functions characterized by high frequency features. However, the performance of MscaleDNNs heavily depends on the parameters in the downing-scaling mapping, which limits their broader application. In this work, we establish a fitting error bound to explain why MscaleDNNs are advantageous for approximating high frequency functions. Building on this insight, we construct a hybrid feature embedding to enhance the accuracy and robustness of the downing-scaling mapping. To reduce the dependency of MscaleDNNs on parameters in the downing-scaling mapping, we propose frequency-adaptive MscaleDNNs, which adaptively adjust these parameters based on a posterior error estimate that captures the frequency information of the fitted functions. Numerical examples, including wave propagation and the propagation of a localized solution of the Schrödinger equation with a smooth potential near the semi-classical limit, are presented. These examples demonstrate that the frequency-adaptive MscaleDNNs improve accuracy by two to three orders of magnitude compared to standard MscaleDNNs.</div></div>","PeriodicalId":55222,"journal":{"name":"Computer Methods in Applied Mechanics and Engineering","volume":"437 ","pages":"Article 117751"},"PeriodicalIF":6.9,"publicationDate":"2025-01-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143071666","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-01-25DOI: 10.1016/j.cma.2025.117742
Nikita Kartashov, Nikolaos N. Vlassis
Microstructure plays a critical role in determining the macroscopic properties of materials, with applications spanning alloy design, MEMS devices, and tissue engineering, among many others. Computational frameworks have been developed to capture the complex relationship between microstructure and material behavior. However, despite these advancements, the steep learning curve associated with domain-specific knowledge and complex algorithms restricts the broader application of these tools. To lower this barrier, we propose a framework that integrates Natural Language Processing (NLP), Large Language Models (LLMs), and Denoising Diffusion Probabilistic Models (DDPMs) to enable microstructure design using intuitive natural language commands. Our framework employs contextual data augmentation, driven by a pretrained LLM, to generate and expand a diverse dataset of microstructure descriptors. A retrained Named Entity Recognition (NER) model extracts relevant microstructure descriptors from user-provided natural language inputs, which are then used by the DDPM to generate microstructures with targeted mechanical properties and topological features. The NLP and DDPM components of the framework are modular, allowing for separate training and validation, which ensures flexibility in adapting the framework to different datasets and use cases. A surrogate model system is employed to rank and filter generated samples based on their alignment with target properties. This work introduces a comprehensive framework that bridges natural language processing and mechanics, addressing key challenges such as the lack of training data, syntax invariance in textual descriptors, and precision in text embeddings. Demonstrated on a database of nonlinear hyperelastic microstructures, this framework serves as a prototype for accessible inverse design of microstructures, starting from intuitive natural language commands.
{"title":"A large language model and denoising diffusion framework for targeted design of microstructures with commands in natural language","authors":"Nikita Kartashov, Nikolaos N. Vlassis","doi":"10.1016/j.cma.2025.117742","DOIUrl":"10.1016/j.cma.2025.117742","url":null,"abstract":"<div><div>Microstructure plays a critical role in determining the macroscopic properties of materials, with applications spanning alloy design, MEMS devices, and tissue engineering, among many others. Computational frameworks have been developed to capture the complex relationship between microstructure and material behavior. However, despite these advancements, the steep learning curve associated with domain-specific knowledge and complex algorithms restricts the broader application of these tools. To lower this barrier, we propose a framework that integrates Natural Language Processing (NLP), Large Language Models (LLMs), and Denoising Diffusion Probabilistic Models (DDPMs) to enable microstructure design using intuitive natural language commands. Our framework employs contextual data augmentation, driven by a pretrained LLM, to generate and expand a diverse dataset of microstructure descriptors. A retrained Named Entity Recognition (NER) model extracts relevant microstructure descriptors from user-provided natural language inputs, which are then used by the DDPM to generate microstructures with targeted mechanical properties and topological features. The NLP and DDPM components of the framework are modular, allowing for separate training and validation, which ensures flexibility in adapting the framework to different datasets and use cases. A surrogate model system is employed to rank and filter generated samples based on their alignment with target properties. This work introduces a comprehensive framework that bridges natural language processing and mechanics, addressing key challenges such as the lack of training data, syntax invariance in textual descriptors, and precision in text embeddings. Demonstrated on a database of nonlinear hyperelastic microstructures, this framework serves as a prototype for accessible inverse design of microstructures, starting from intuitive natural language commands.</div></div>","PeriodicalId":55222,"journal":{"name":"Computer Methods in Applied Mechanics and Engineering","volume":"437 ","pages":"Article 117742"},"PeriodicalIF":6.9,"publicationDate":"2025-01-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143071667","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We propose a superconvergent isogeometric collocation (IGSC) method based on quartic -continuous box splines on triangular partitions. By leveraging the superconvergence characteristics of box splines, we identify several sets of desirable collocation points. Numerical experiments demonstrate that the isogeometric collocation utilizing these collocation points achieves convergence rates comparable to those of isogeometric Galerkin methods in terms of the and -norms. The results further reveal that, while face centroids are suboptimal as collocation points for box splines, edge midpoints are effective for IGSC. The proposed approach is tested on Poisson equations and linear elasticity problems, making comparisons with the isogeometric Galerkin method. Additionally, a thorough comparison of the computational costs between the proposed technique and isogeometric Galerkin methods is presented.
{"title":"Superconvergent isogeometric collocation with box splines","authors":"Hailun Xu , Hongmei Kang , Falai Chen , Zhimin Zhang","doi":"10.1016/j.cma.2025.117763","DOIUrl":"10.1016/j.cma.2025.117763","url":null,"abstract":"<div><div>We propose a superconvergent isogeometric collocation (IGSC) method based on quartic <span><math><msup><mrow><mi>C</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span>-continuous box splines on triangular partitions. By leveraging the superconvergence characteristics of box splines, we identify several sets of desirable collocation points. Numerical experiments demonstrate that the isogeometric collocation utilizing these collocation points achieves convergence rates comparable to those of isogeometric Galerkin methods in terms of the <span><math><msub><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span> and <span><math><msub><mrow><mi>H</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span>-norms. The results further reveal that, while face centroids are suboptimal as collocation points for box splines, edge midpoints are effective for IGSC. The proposed approach is tested on Poisson equations and linear elasticity problems, making comparisons with the isogeometric Galerkin method. Additionally, a thorough comparison of the computational costs between the proposed technique and isogeometric Galerkin methods is presented.</div></div>","PeriodicalId":55222,"journal":{"name":"Computer Methods in Applied Mechanics and Engineering","volume":"437 ","pages":"Article 117763"},"PeriodicalIF":6.9,"publicationDate":"2025-01-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143100537","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-01-23DOI: 10.1016/j.cma.2025.117730
Markus Schewe , Isabelle Noll , Thorsten Bartel , Andreas Menzel
The present paper establishes a simulation framework for modelling the deposition and solidification of steel melt in Directed Energy Deposition with a Laser Beam (DED-LB) by using the Particle Finite Element Method (PFEM). Unlike traditional finite element methods, the remeshing framework makes it possible to resolve the interaction between molten metal and substrate upon deposition, solidification and cooling, which provides a framework for accurately predicting residual stresses and distortion in the final part. The material model incorporates a liquid–solid phase transformation described by phase fractions, allowing for a precise definition of transformation stretches, latent heat and fundamental changes in the constitutive behaviour, whereas a purely temperature dependent phase evolution keeps the numerical cost manageable. While focusing on a two-dimensional (2d) simulation for simplicity and observability of the mesh adaptation, the methodology is extensible to a 3d setting. Key advancements include refined remeshing techniques of the connection zone and a large strain melt and solidification material model. The simulation results demonstrate the potential of the proposed framework for capturing critical aspects of DED-LB processes, laying the basis for extensive process simulations.
{"title":"Towards the simulation of metal deposition with the Particle Finite Element Method and a phase transformation model","authors":"Markus Schewe , Isabelle Noll , Thorsten Bartel , Andreas Menzel","doi":"10.1016/j.cma.2025.117730","DOIUrl":"10.1016/j.cma.2025.117730","url":null,"abstract":"<div><div>The present paper establishes a simulation framework for modelling the deposition and solidification of steel melt in Directed Energy Deposition with a Laser Beam (DED-LB) by using the Particle Finite Element Method (PFEM). Unlike traditional finite element methods, the remeshing framework makes it possible to resolve the interaction between molten metal and substrate upon deposition, solidification and cooling, which provides a framework for accurately predicting residual stresses and distortion in the final part. The material model incorporates a liquid–solid phase transformation described by phase fractions, allowing for a precise definition of transformation stretches, latent heat and fundamental changes in the constitutive behaviour, whereas a purely temperature dependent phase evolution keeps the numerical cost manageable. While focusing on a two-dimensional (2d) simulation for simplicity and observability of the mesh adaptation, the methodology is extensible to a 3d setting. Key advancements include refined remeshing techniques of the connection zone and a large strain melt and solidification material model. The simulation results demonstrate the potential of the proposed framework for capturing critical aspects of DED-LB processes, laying the basis for extensive process simulations.</div></div>","PeriodicalId":55222,"journal":{"name":"Computer Methods in Applied Mechanics and Engineering","volume":"437 ","pages":"Article 117730"},"PeriodicalIF":6.9,"publicationDate":"2025-01-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143027313","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-01-23DOI: 10.1016/j.cma.2025.117762
Yan Shang , Ming Sun , Song Cen , Chen-Feng Li
The evaluation of flexoelectric composites with architected microstructures requires a reasonable estimation of their effective properties. To accomplish this, a computational homogenization scheme for flexoelectric composites based on the consistent couple stress theory is proposed in this work, where the extended Hill's lemma is strictly established and accordingly, different types of admissible boundary conditions, including the periodic boundary condition, required to impose on the representative volume element are systematically derived from the Hill macrohomogeneity condition. In particular, in order to show more clearly how to deduce the effective constitutive coefficients via the proposed method, its implementation in the plane problem is described in detail. Finally, to verify the effectiveness of the method, numerical examples are examined in which the computations are carried out by using the penalty 8-node quadrilateral element developed following the unsymmetric finite element method. The numerical results fully prove that the proposed method can estimate the equivalent properties of flexoelectric composites very effectively.
{"title":"Computational homogenization of flexoelectric composites within the consistent couple stress theory","authors":"Yan Shang , Ming Sun , Song Cen , Chen-Feng Li","doi":"10.1016/j.cma.2025.117762","DOIUrl":"10.1016/j.cma.2025.117762","url":null,"abstract":"<div><div>The evaluation of flexoelectric composites with architected microstructures requires a reasonable estimation of their effective properties. To accomplish this, a computational homogenization scheme for flexoelectric composites based on the consistent couple stress theory is proposed in this work, where the extended Hill's lemma is strictly established and accordingly, different types of admissible boundary conditions, including the periodic boundary condition, required to impose on the representative volume element are systematically derived from the Hill macrohomogeneity condition. In particular, in order to show more clearly how to deduce the effective constitutive coefficients via the proposed method, its implementation in the plane problem is described in detail. Finally, to verify the effectiveness of the method, numerical examples are examined in which the computations are carried out by using the penalty 8-node quadrilateral element developed following the unsymmetric finite element method. The numerical results fully prove that the proposed method can estimate the equivalent properties of flexoelectric composites very effectively.</div></div>","PeriodicalId":55222,"journal":{"name":"Computer Methods in Applied Mechanics and Engineering","volume":"437 ","pages":"Article 117762"},"PeriodicalIF":6.9,"publicationDate":"2025-01-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143027301","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Although parallel active learning reliability analysis is promising and has been widely studied, there remains an open question regarding how to achieve better theoretical consistency and avoid reliance on empirical practices heavily. A new parallel Bayesian active learning reliability method is developed in this study. First, in Bayesian failure probability estimation, a metric called integrated probability of misclassification (IPM) is defined from the upper bound of mean absolute deviation of failure probability. Then, a multi-point learning function called -point integrated probability of misclassification reduction (-IPMR) is proposed to guide the selection of a batch of new samples that maximize the expected reduction of IPM. To further reduce the computational overhead, the fast -IPMR-guided parallel Bayesian active learning reliability analysis is conducted through four key workarounds. (i) The -IPMR is substituted by its theoretically analogous but computationally cheaper variant. (ii) A stepwise maximization of -IPMR is deployed to replace the cumbersome direct maximization approach. (iii) The number of new samples added per iteration is identified in an adaptive manner. (iv) A hybrid convergence criterion is specified according to the actual reduction of IPM at each iteration. Owing to the core role of IPM, we fuse the three major ingredients, i.e., Bayesian inference of failure probability, multi-point enrichment process, and convergence criterion, in a theoretically consistent way. The performance of the proposed method is testified on four examples of varying complexity. The results indicate that the proposed approach needs a fewer number of iterations than those existing ones and thus is more computationally efficient, particularly when dealing with time-intensive complex reliability problems.
{"title":"A theoretically-consistent parallel enrichment strategy for Bayesian active learning reliability analysis","authors":"Tong Zhou , Tong Guo , Xujia Zhu , Masaru Kitahara , Jize Zhang","doi":"10.1016/j.cma.2025.117752","DOIUrl":"10.1016/j.cma.2025.117752","url":null,"abstract":"<div><div>Although parallel active learning reliability analysis is promising and has been widely studied, there remains an open question regarding how to achieve better theoretical consistency and avoid reliance on empirical practices heavily. A new parallel Bayesian active learning reliability method is developed in this study. First, in Bayesian failure probability estimation, a metric called integrated probability of misclassification (IPM) is defined from the upper bound of mean absolute deviation of failure probability. Then, a multi-point learning function called <span><math><mi>k</mi></math></span>-point integrated probability of misclassification reduction (<span><math><mi>k</mi></math></span>-IPMR) is proposed to guide the selection of a batch of <span><math><mrow><mi>k</mi><mrow><mo>(</mo><mo>≥</mo><mn>1</mn><mo>)</mo></mrow></mrow></math></span> new samples that maximize the expected reduction of IPM. To further reduce the computational overhead, the fast <span><math><mi>k</mi></math></span>-IPMR-guided parallel Bayesian active learning reliability analysis is conducted through four key workarounds. (i) The <span><math><mi>k</mi></math></span>-IPMR is substituted by its theoretically analogous but computationally cheaper variant. (ii) A stepwise maximization of <span><math><mi>k</mi></math></span>-IPMR is deployed to replace the cumbersome direct maximization approach. (iii) The number of new samples added per iteration is identified in an adaptive manner. (iv) A hybrid convergence criterion is specified according to the actual reduction of IPM at each iteration. Owing to the core role of IPM, we fuse the three major ingredients, i.e., Bayesian inference of failure probability, multi-point enrichment process, and convergence criterion, in a theoretically consistent way. The performance of the proposed method is testified on four examples of varying complexity. The results indicate that the proposed approach needs a fewer number of iterations than those existing ones and thus is more computationally efficient, particularly when dealing with time-intensive complex reliability problems.</div></div>","PeriodicalId":55222,"journal":{"name":"Computer Methods in Applied Mechanics and Engineering","volume":"437 ","pages":"Article 117752"},"PeriodicalIF":6.9,"publicationDate":"2025-01-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143027317","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-01-22DOI: 10.1016/j.cma.2025.117764
Akshay J. Thomas , Ilias Bilionis , Eduardo Barocio , R. Byron Pipes
We introduce a new method for solving parametric heat transfer partial differential equations on evolving geometries in advanced manufacturing applications. Physics-informed neural networks (PINNs) are a popular framework for integrating experimental data with known physical laws specified via partial differential equations (PDEs). Despite their increasing popularity, applying PINNs to manufacturing problems is limited compared to fluid and solid mechanics problems. The applications of PINNs acting as PDE solvers are absent where material is being added or removed. The objective of our work is to address this gap. By proposing a new loss function, we aim to expand the applications of PINNs for heat transfer to manufacturing problems with evolving geometries. Our method obviates the need for mesh-based discretization and time-marching schemes for evolving geometries. We consider predicting the transient temperature history in additive manufacturing as a single bead of material is deposited. We consider various evolving mixed Dirichlet and Neumann boundary condition cases to test our methodology. We verify our methodology by comparing our results with a validated finite element (FE) solver and observe that the results are in excellent agreement. Our method is naturally biased to respect causality, achieved by an automatic decrease in collocation point density as the geometry evolves. We extend our method to solve a parametric heat transfer equation for the single bead addition problem and outline the advantages in computational cost provided by our parametric solver compared to running multiple instances of an FE solver.
{"title":"Causality enforcing parametric heat transfer solvers for evolving geometries in advanced manufacturing","authors":"Akshay J. Thomas , Ilias Bilionis , Eduardo Barocio , R. Byron Pipes","doi":"10.1016/j.cma.2025.117764","DOIUrl":"10.1016/j.cma.2025.117764","url":null,"abstract":"<div><div>We introduce a new method for solving parametric heat transfer partial differential equations on evolving geometries in advanced manufacturing applications. Physics-informed neural networks (PINNs) are a popular framework for integrating experimental data with known physical laws specified via partial differential equations (PDEs). Despite their increasing popularity, applying PINNs to manufacturing problems is limited compared to fluid and solid mechanics problems. The applications of PINNs acting as PDE solvers are absent where material is being added or removed. The objective of our work is to address this gap. By proposing a new loss function, we aim to expand the applications of PINNs for heat transfer to manufacturing problems with evolving geometries. Our method obviates the need for mesh-based discretization and time-marching schemes for evolving geometries. We consider predicting the transient temperature history in additive manufacturing as a single bead of material is deposited. We consider various evolving mixed Dirichlet and Neumann boundary condition cases to test our methodology. We verify our methodology by comparing our results with a validated finite element (FE) solver and observe that the results are in excellent agreement. Our method is naturally biased to respect causality, achieved by an automatic decrease in collocation point density as the geometry evolves. We extend our method to solve a parametric heat transfer equation for the single bead addition problem and outline the advantages in computational cost provided by our parametric solver compared to running multiple instances of an FE solver.</div></div>","PeriodicalId":55222,"journal":{"name":"Computer Methods in Applied Mechanics and Engineering","volume":"437 ","pages":"Article 117764"},"PeriodicalIF":6.9,"publicationDate":"2025-01-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143027318","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-01-22DOI: 10.1016/j.cma.2025.117778
Yichen Wu , Lei Wang , Zeshang Li , Lianmei Wu , Yaru Liu
The increasing demand for load-carrying multiscale structures with ultimate lightness requires corresponding development in topology optimization methods. However, current multiscale topology optimization methods are hindered by the contradiction between the freedom of design space and the dimensionality of the design variables. Moreover, the unstable additive manufacturing process and working conditions would result in possible structural failure of the multiscale structure optimized under deterministic conditions. To address these problems, we propose a deep generative multiscale topology optimization framework considering both manufacturing defects and parametrical uncertainties. A database consisting of minimum volume unit cell topologies is obtained via the inverse homogenization method. Then the variational autoencoder network is introduced to capture the patterns in the database and to reconstruct unit cells with a low-dimensional latent vector, which effectively compresses the number of design variables for a microstructure. Then, a self-adaptive clustering strategy is proposed to efficiently quantify the influence of random manufacturing defects in microstructures. A reliability-based optimization framework is constructed with a reliability index to evaluate the complex effect of multisource uncertainties. The effectiveness of the proposed framework is validated through a series of numerical examples, and conclusions are presented at the end of the article.
{"title":"A deep generative multiscale topology optimization framework considering manufacturing defects and parametrical uncertainties","authors":"Yichen Wu , Lei Wang , Zeshang Li , Lianmei Wu , Yaru Liu","doi":"10.1016/j.cma.2025.117778","DOIUrl":"10.1016/j.cma.2025.117778","url":null,"abstract":"<div><div>The increasing demand for load-carrying multiscale structures with ultimate lightness requires corresponding development in topology optimization methods. However, current multiscale topology optimization methods are hindered by the contradiction between the freedom of design space and the dimensionality of the design variables. Moreover, the unstable additive manufacturing process and working conditions would result in possible structural failure of the multiscale structure optimized under deterministic conditions. To address these problems, we propose a deep generative multiscale topology optimization framework considering both manufacturing defects and parametrical uncertainties. A database consisting of minimum volume unit cell topologies is obtained via the inverse homogenization method. Then the variational autoencoder network is introduced to capture the patterns in the database and to reconstruct unit cells with a low-dimensional latent vector, which effectively compresses the number of design variables for a microstructure. Then, a self-adaptive clustering strategy is proposed to efficiently quantify the influence of random manufacturing defects in microstructures. A reliability-based optimization framework is constructed with a reliability index to evaluate the complex effect of multisource uncertainties. The effectiveness of the proposed framework is validated through a series of numerical examples, and conclusions are presented at the end of the article.</div></div>","PeriodicalId":55222,"journal":{"name":"Computer Methods in Applied Mechanics and Engineering","volume":"437 ","pages":"Article 117778"},"PeriodicalIF":6.9,"publicationDate":"2025-01-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143027315","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-01-22DOI: 10.1016/j.cma.2025.117755
Xi Wang , Zhen-Yu Yin , Wei Wu , He-Hua Zhu
Physics-informed neural networks (PINNs) are well-regarded for their capabilities in inverse analysis. However, efficient convergence is hard to achieve due to the necessity of simultaneously handling physics constraints, data constraints, blackbox weights, and blackbox biases. Consequently, PINNs are highly challenged in the inverse analysis of unknown boundary loadings and heterogeneous material parameters, particularly for three-dimensional engineering structures. To address these limitations, this study develops a novel differentiable finite element method (DFEM) based on Galerkin discretization for diverse inverse analysis. The proposed DFEM directly embeds the weak form of the partial differential equation into a discretized and differentiable computational graph, yielding a loss function from fully interpretable trainable parameters. Moreover, the labeled data, including boundary conditions, are strictly encoded into the computational graph without additional training. Finally, two benchmarks validate the DFEM's superior efficiency and accuracy: (1) With only 0.3 % training iterations, the DFEM can achieve an accuracy three orders of magnitude higher for the inverse analysis of unknown loadings. (2) With a training time five orders of magnitude faster, the DFEM is validated to be five orders of magnitude more accurate in determining unknown material parameters. Furthermore, two cases validate DFEM as effective for three-dimensional engineering structures: (1) A damaged cantilever beam characterized by twenty heterogeneous materials with forty unknown parameters is efficiently solved. (2) A tunnel lining ring with sparse noisy data under unknown heterogeneous boundary loadings is successfully analyzed. These problems are solved in seconds, corroborating DFEM's potential for engineering applications. Additionally, the DFEM framework can be generalized to a Physics-Encoded Numerical Network (PENN) for further development and exploration.
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