This paper proposes and studies two extensions of applying hp-variational physics-informed neural networks, more precisely the FastVPINNs framework, to convection-dominated convection–diffusion–reaction problems. First, a term in the spirit of a SUPG stabilization is included in the loss functional and a network architecture is proposed that predicts spatially varying stabilization parameters. Having observed that the selection of the indicator function in hard-constrained Dirichlet boundary conditions has a big impact on the accuracy of the computed solutions, the second novelty is the proposal of a network architecture that learns good parameters for a class of indicator functions. Numerical studies show that both proposals lead to noticeably more accurate results than approaches that can be found in the literature.
{"title":"Improving hp-variational physics-informed neural networks for steady-state convection-dominated problems","authors":"Thivin Anandh , Divij Ghose , Himanshu Jain , Pratham Sunkad , Sashikumaar Ganesan , Volker John","doi":"10.1016/j.cma.2025.117797","DOIUrl":"10.1016/j.cma.2025.117797","url":null,"abstract":"<div><div>This paper proposes and studies two extensions of applying hp-variational physics-informed neural networks, more precisely the FastVPINNs framework, to convection-dominated convection–diffusion–reaction problems. First, a term in the spirit of a SUPG stabilization is included in the loss functional and a network architecture is proposed that predicts spatially varying stabilization parameters. Having observed that the selection of the indicator function in hard-constrained Dirichlet boundary conditions has a big impact on the accuracy of the computed solutions, the second novelty is the proposal of a network architecture that learns good parameters for a class of indicator functions. Numerical studies show that both proposals lead to noticeably more accurate results than approaches that can be found in the literature.</div></div>","PeriodicalId":55222,"journal":{"name":"Computer Methods in Applied Mechanics and Engineering","volume":"438 ","pages":"Article 117797"},"PeriodicalIF":6.9,"publicationDate":"2025-02-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143386407","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}
A finite-element method dependent adjoint-based procedure to determine the temperature field of structures based on measured displacements/strains and a set of standard loads is developed and tested. Given a series of force and deformation measurements, the temperature field is obtained by minimizing the adequately weighted differences between the measured and computed values. Three numerical examples — a Plate With a Hole, a Bridge, and a Hoover Dam example — each with multiple sensors distributed in different configurations, demonstrate the procedure’s capabilities. A target temperature distribution is prescribed in all cases, and the displacement sensor data is recorded. The optimization algorithm (here, steepest descent with Barzilai–Borwein step) uses this data to optimize the temperatures such that the same deformation is obtained at the sensor locations. Vertex Morphing is used as a filter to mitigate the ill-conditioning. Results show that the proposed approach can accurately reconstruct the target thermal distribution, especially when more sensors are used. Additionally, it is observed that the sensors do not need to be positioned in the region of interest; the method remains effective as long as the sensors can detect changes related to that area. A comparison with standard spatial interpolation techniques, namely, k-nearest neighbors and ordinary and universal kriging, is performed using temperature sensors in the same configurations. The proposed approach performs remarkably better than the interpolation techniques with a reduction in the norm of up to 41.3%, 93.9%, and 41.3%, for the Plate With a Hole, the Bridge, and the Dam examples, respectively.
{"title":"Adjoint-based recovery of thermal fields from displacement or strain measurements","authors":"Talhah Shamshad Ali Ansari , Rainald Löhner , Roland Wüchner , Harbir Antil , Suneth Warnakulasuriya , Ihar Antonau , Facundo Airaudo","doi":"10.1016/j.cma.2025.117818","DOIUrl":"10.1016/j.cma.2025.117818","url":null,"abstract":"<div><div>A finite-element method dependent adjoint-based procedure to determine the temperature field of structures based on measured displacements/strains and a set of standard loads is developed and tested. Given a series of force and deformation measurements, the temperature field is obtained by minimizing the adequately weighted differences between the measured and computed values. Three numerical examples — a Plate With a Hole, a Bridge, and a Hoover Dam example — each with multiple sensors distributed in different configurations, demonstrate the procedure’s capabilities. A target temperature distribution is prescribed in all cases, and the displacement sensor data is recorded. The optimization algorithm (here, steepest descent with Barzilai–Borwein step) uses this data to optimize the temperatures such that the same deformation is obtained at the sensor locations. Vertex Morphing is used as a filter to mitigate the ill-conditioning. Results show that the proposed approach can accurately reconstruct the target thermal distribution, especially when more sensors are used. Additionally, it is observed that the sensors do not need to be positioned in the region of interest; the method remains effective as long as the sensors can detect changes related to that area. A comparison with standard spatial interpolation techniques, namely, k-nearest neighbors and ordinary and universal kriging, is performed using temperature sensors in the same configurations. The proposed approach performs remarkably better than the interpolation techniques with a reduction in the <span><math><mrow><mi>L</mi><mn>2</mn></mrow></math></span> norm of up to 41.3%, 93.9%, and 41.3%, for the Plate With a Hole, the Bridge, and the Dam examples, respectively.</div></div>","PeriodicalId":55222,"journal":{"name":"Computer Methods in Applied Mechanics and Engineering","volume":"438 ","pages":"Article 117818"},"PeriodicalIF":6.9,"publicationDate":"2025-02-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143386410","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 : 2025-02-12DOI: 10.1016/j.cma.2025.117807
R. Fu , D. Xiao , A.G. Buchan , X. Lin , Y. Feng , G. Dong
Reduced order modelling is popular and widely used in engineering as it has the potential to gain several orders of magnitude CPU speedup for simulations with different parameters such as different initial or boundary conditions. This work presents a new parametric non-linear non-intrusive reduced-order model (P-NLNIROM) for the fluid problems, which extends the capabilities for non-linear NIROM Fuet al. (2023) on parametric problems. The Deep Auto-Encoder (DAE), Deep Residual Learning Neural network (ResNet), and deep transfer learning are used to construct the P-NLNIROM. The DAE is developed to project or map the original high-dimensional dynamical systems into a much lower dimensional nonlinear reduced latent space and the deep ResNet is used to construct a set of functions that represents the relationships between model input parameters (such as initial or boundary conditions) and extracted representations in the latent or reduced space (reduced representation of fluid dynamics). Transfer learning is used to extend the predictive capability of the model for different parameters. The novelty of this work lies in that ResNet and transfer learning are used to predict different parametric conditions for the AutoEncoder-based, non-linear, non-intrusive reduced-order model (NLNIROM). Transfer learning expands the predictive range of parametric space and makes the transferred P-NLNIROM perform well with much less data. The capability of this new P-NLNIROM is illustrated numerically by two test cases: a lock exchange, and a flow past a cylinder. The results obtained show that the P-NLNIROM performs well and the transferred model shows more promising results under new parameter conditions.
{"title":"A parametric non-linear non-intrusive reduce-order model using deep transfer learning","authors":"R. Fu , D. Xiao , A.G. Buchan , X. Lin , Y. Feng , G. Dong","doi":"10.1016/j.cma.2025.117807","DOIUrl":"10.1016/j.cma.2025.117807","url":null,"abstract":"<div><div>Reduced order modelling is popular and widely used in engineering as it has the potential to gain several orders of magnitude CPU speedup for simulations with different parameters such as different initial or boundary conditions. This work presents a new parametric non-linear non-intrusive reduced-order model (P-NLNIROM) for the fluid problems, which extends the capabilities for non-linear NIROM Fuet al. (2023) on parametric problems. The Deep Auto-Encoder (DAE), Deep Residual Learning Neural network (ResNet), and deep transfer learning are used to construct the P-NLNIROM. The DAE is developed to project or map the original high-dimensional dynamical systems into a much lower dimensional nonlinear reduced latent space and the deep ResNet is used to construct a set of functions that represents the relationships between model input parameters (such as initial or boundary conditions) and extracted representations in the latent or reduced space (reduced representation of fluid dynamics). Transfer learning is used to extend the predictive capability of the model for different parameters. The novelty of this work lies in that ResNet and transfer learning are used to predict different parametric conditions for the AutoEncoder-based, non-linear, non-intrusive reduced-order model (NLNIROM). Transfer learning expands the predictive range of parametric space and makes the transferred P-NLNIROM perform well with much less data. The capability of this new P-NLNIROM is illustrated numerically by two test cases: a lock exchange, and a flow past a cylinder. The results obtained show that the P-NLNIROM performs well and the transferred model shows more promising results under new parameter conditions.</div></div>","PeriodicalId":55222,"journal":{"name":"Computer Methods in Applied Mechanics and Engineering","volume":"438 ","pages":"Article 117807"},"PeriodicalIF":6.9,"publicationDate":"2025-02-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143386411","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 : 2025-02-11DOI: 10.1016/j.cma.2025.117816
Francesco Duronio , Andrea Di Mascio
Many fluid flows of engineering interest involve high-density ratios, the formation of shock waves, and the presence of multiple chemical species. In these situations, obtaining accurate results depends on the computational fluid dynamics (CFD) algorithm’s ability to ensure stable and robust time and space discretization of the governing equations, effectively minimizing numerical diffusion.
In this work, we develop and evaluate a low-dissipation, high-order solver for the Navier–Stokes equations applicable to compressible, multi-species flows within the OpenFOAM library. We introduce a new shock sensor specifically designed for high-order discretization methods, which guarantees accuracy up to the fourth order in both space and time when using unstructured grids. Additionally, this approach is compatible with real-gas equations of state.
We evaluated the solver performances on three test cases, each chosen for its relevance to real-world engineering problems: a 1D multi-specie shock tube, a 2D shock tube where Richtmyer–Meshkov instability develops as a consequence of bubble-shock interaction, and, finally, the flow through the complex 3D geometry of a high-pressure fuel injector used in propulsion applications, to investigate its performances on unstructured grids. The results confirm the effectiveness of the proposed sensor in scenarios characterized by field discontinuities, shock waves, high-density ratio, and distorted grids. This means that our solver can accurately simulate complex fluid flows in engineering applications where these conditions are often encountered.
The developed high-order scheme and shock sensor were implemented in an OpenFoam solver called rhoCubic4kFoam.
{"title":"Development of an efficient shock sensor for high-order multi-species compressible flow solvers on unstructured grids","authors":"Francesco Duronio , Andrea Di Mascio","doi":"10.1016/j.cma.2025.117816","DOIUrl":"10.1016/j.cma.2025.117816","url":null,"abstract":"<div><div>Many fluid flows of engineering interest involve high-density ratios, the formation of shock waves, and the presence of multiple chemical species. In these situations, obtaining accurate results depends on the computational fluid dynamics (CFD) algorithm’s ability to ensure stable and robust time and space discretization of the governing equations, effectively minimizing numerical diffusion.</div><div>In this work, we develop and evaluate a low-dissipation, high-order solver for the Navier–Stokes equations applicable to compressible, multi-species flows within the OpenFOAM library. We introduce a new shock sensor specifically designed for high-order discretization methods, which guarantees accuracy up to the fourth order in both space and time when using unstructured grids. Additionally, this approach is compatible with real-gas equations of state.</div><div>We evaluated the solver performances on three test cases, each chosen for its relevance to real-world engineering problems: a 1D multi-specie shock tube, a 2D shock tube where Richtmyer–Meshkov instability develops as a consequence of bubble-shock interaction, and, finally, the flow through the complex 3D geometry of a high-pressure fuel injector used in propulsion applications, to investigate its performances on unstructured grids. The results confirm the effectiveness of the proposed sensor in scenarios characterized by field discontinuities, shock waves, high-density ratio, and distorted grids. This means that our solver can accurately simulate complex fluid flows in engineering applications where these conditions are often encountered.</div><div>The developed high-order scheme and shock sensor were implemented in an OpenFoam solver called <em>rhoCubic4kFoam</em>.</div></div>","PeriodicalId":55222,"journal":{"name":"Computer Methods in Applied Mechanics and Engineering","volume":"438 ","pages":"Article 117816"},"PeriodicalIF":6.9,"publicationDate":"2025-02-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143386404","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 : 2025-02-11DOI: 10.1016/j.cma.2025.117811
Maria Strazzullo , Francesco Ballarin , Traian Iliescu , Tomás Chacón Rebollo
The evolve-filter (EF) model is a filter-based numerical stabilization for under-resolved convection-dominated flows. EF is a simple, modular, and effective strategy for both full-order models (FOMs) and reduced-order models (ROMs). It is well-known, however, that when the filter radius is too large, EF can be overdiffusive and yield inaccurate results. To alleviate this, EF is usually supplemented with a relaxation step. The relaxation parameter, however, is very sensitive with respect to the model parameters. In this paper, we propose a novel strategy to alleviate the EF overdiffusivity. Specifically, we leverage the variational multiscale (VMS) framework to separate the large resolved scales from the small resolved scales in the evolved velocity, and we use the filtered small scales to correct the large scales. Furthermore, in the new VMS-EF strategy, we use two different approaches to decompose the evolved velocity: the VMS Evolve-Filter-Filter-Correct (VMS-EFFC) and the VMS Evolve-Postprocess-Filter-Correct (VMS-EPFC) algorithms. The new VMS-based algorithms yield significantly more accurate results than the standard EF in both the FOM and the ROM simulations of a flow past a cylinder at Reynolds number Re = 1000.
{"title":"Variational multiscale evolve and filter strategies for convection-dominated flows","authors":"Maria Strazzullo , Francesco Ballarin , Traian Iliescu , Tomás Chacón Rebollo","doi":"10.1016/j.cma.2025.117811","DOIUrl":"10.1016/j.cma.2025.117811","url":null,"abstract":"<div><div>The evolve-filter (EF) model is a filter-based numerical stabilization for under-resolved convection-dominated flows. EF is a simple, modular, and effective strategy for both full-order models (FOMs) and reduced-order models (ROMs). It is well-known, however, that when the filter radius is too large, EF can be overdiffusive and yield inaccurate results. To alleviate this, EF is usually supplemented with a relaxation step. The relaxation parameter, however, is very sensitive with respect to the model parameters. In this paper, we propose a novel strategy to alleviate the EF overdiffusivity. Specifically, we leverage the variational multiscale (VMS) framework to separate the large resolved scales from the small resolved scales in the evolved velocity, and we use the filtered small scales to correct the large scales. Furthermore, in the new VMS-EF strategy, we use two different approaches to decompose the evolved velocity: the VMS Evolve-Filter-Filter-Correct (VMS-EFFC) and the VMS Evolve-Postprocess-Filter-Correct (VMS-EPFC) algorithms. The new VMS-based algorithms yield significantly more accurate results than the standard EF in both the FOM and the ROM simulations of a flow past a cylinder at Reynolds number Re = 1000.</div></div>","PeriodicalId":55222,"journal":{"name":"Computer Methods in Applied Mechanics and Engineering","volume":"438 ","pages":"Article 117811"},"PeriodicalIF":6.9,"publicationDate":"2025-02-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143386403","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 : 2025-02-11DOI: 10.1016/j.cma.2025.117814
Chen Yang , Qianqian Yu
In force reconstruction, multi-source incomplete information makes it difficult for traditional methods to model and solve the problem accurately, especially with noise or measurement errors. Inspired by multi-objective optimization, this paper proposes a novel set theory-based regularization approach (STR) to enhance adaptability to uncertainties and improve reconstruction accuracy and robustness. The nominal force inversion is constituted and then extended into the interval uncertainty framework, and an effective orthogonal sampling-based interval prediction method is proposed to analyze the coupled effect of force inversion and uncertainty propagation. Once the uncertainty level of the complex structure is known, this prediction method can accurately and quickly estimate the fluctuation bound of the identified force. Enlightened by the completely same logic between the regularization method used in force inversion and multi-objective optimization problem, namely, simultaneously satisfying the norm minimization of the solution and the residual parameter, this study develops a novel multi-objective optimization-inspired set theory-based regularization parameter selection method. This method incorporates the interval dominance relationship to select the most competitive regularization parameter under interval uncertainties. Therefore, an accurate reconstuction framework with bounded uncertainties is finally proposed and verified by two numerical examples.
{"title":"Multi-objective optimization-inspired set theory-based regularization approach for force reconstruction with bounded uncertainties","authors":"Chen Yang , Qianqian Yu","doi":"10.1016/j.cma.2025.117814","DOIUrl":"10.1016/j.cma.2025.117814","url":null,"abstract":"<div><div>In force reconstruction, multi-source incomplete information makes it difficult for traditional methods to model and solve the problem accurately, especially with noise or measurement errors. Inspired by multi-objective optimization, this paper proposes a novel set theory-based regularization approach (STR) to enhance adaptability to uncertainties and improve reconstruction accuracy and robustness. The nominal force inversion is constituted and then extended into the interval uncertainty framework, and an effective orthogonal sampling-based interval prediction method is proposed to analyze the coupled effect of force inversion and uncertainty propagation. Once the uncertainty level of the complex structure is known, this prediction method can accurately and quickly estimate the fluctuation bound of the identified force. Enlightened by the completely same logic between the regularization method used in force inversion and multi-objective optimization problem, namely, simultaneously satisfying the norm minimization of the solution and the residual parameter, this study develops a novel multi-objective optimization-inspired set theory-based regularization parameter selection method. This method incorporates the interval dominance relationship to select the most competitive regularization parameter under interval uncertainties. Therefore, an accurate reconstuction framework with bounded uncertainties is finally proposed and verified by two numerical examples.</div></div>","PeriodicalId":55222,"journal":{"name":"Computer Methods in Applied Mechanics and Engineering","volume":"438 ","pages":"Article 117814"},"PeriodicalIF":6.9,"publicationDate":"2025-02-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143386401","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 : 2025-02-09DOI: 10.1016/j.cma.2025.117825
Mohamed Abdel-Basset , Reda Mohamed , Mohamed Abouhawwash
This study presents a new nature-inspired metaheuristic algorithm, known as the fungal growth optimizer (FGO), which is inspired by fungal growth behavior in nature. Fungal growth behavior includes hyphal growth, branching, and spore germination. Hyphal growth behavior replicates hyphal extension and chemotropism to precisely explore the search space, and thus, reach and exploit nutrient-rich regions. This behavior provides a variety of search patterns in FGO, promoting its performance against stagnation into local optima and slow convergence speed. The branching behavior replicates how new hyphal branches from the side of an existing hypha explore the surrounding regions in search of more nutrients, promoting the exploratory operator throughout the optimization process. The final behavior is spore germination, which represents how existing hyphae explore new environments to reach safer and nutrient-rich areas. When spores land in an environment that is rich in moisture and nutrition, they germinate and grow. FGO assumes that spores will land in a random position at the beginning of the optimization process to promote the exploratory operator. As the optimization process is exceeded, this random position is transformed into a position between the best-so-far solution and a random position, promoting the exploitation operator while preventing premature convergence. FGO is evaluated against four well-known Congress on Evolutionary Computation (CEC) benchmarks (CEC2020, CEC2017, CEC2014, and CEC2022) and eleven engineering design problems. In addition, it is compared with fifteen recently proposed algorithms and eleven highly-performing algorithms, such as L-SHADE, LSHADE-cnEpSin, AL-SHADE, mantis search algorithm (MSA), IMODE, AGSK, SOMA_T3A, HyDE-DF, modified LSHADE-SPACMA, SHADE, and LSHADE-SPACMA, to demonstrate its superiority. According to the experimental results, FGO outperforms or is competitive with all of the compared algorithms for the majority of the test functions, implying that it is a high-performing optimizer and a powerful alternative technique for dealing with complex optimization problems. The FGO source code is available on this link
{"title":"Fungal growth optimizer: A novel nature-inspired metaheuristic algorithm for stochastic optimization","authors":"Mohamed Abdel-Basset , Reda Mohamed , Mohamed Abouhawwash","doi":"10.1016/j.cma.2025.117825","DOIUrl":"10.1016/j.cma.2025.117825","url":null,"abstract":"<div><div>This study presents a new nature-inspired metaheuristic algorithm, known as the fungal growth optimizer (FGO), which is inspired by fungal growth behavior in nature. Fungal growth behavior includes hyphal growth, branching, and spore germination. Hyphal growth behavior replicates hyphal extension and chemotropism to precisely explore the search space, and thus, reach and exploit nutrient-rich regions. This behavior provides a variety of search patterns in FGO, promoting its performance against stagnation into local optima and slow convergence speed. The branching behavior replicates how new hyphal branches from the side of an existing hypha explore the surrounding regions in search of more nutrients, promoting the exploratory operator throughout the optimization process. The final behavior is spore germination, which represents how existing hyphae explore new environments to reach safer and nutrient-rich areas. When spores land in an environment that is rich in moisture and nutrition, they germinate and grow. FGO assumes that spores will land in a random position at the beginning of the optimization process to promote the exploratory operator. As the optimization process is exceeded, this random position is transformed into a position between the best-so-far solution and a random position, promoting the exploitation operator while preventing premature convergence. FGO is evaluated against four well-known Congress on Evolutionary Computation (CEC) benchmarks (CEC2020, CEC2017, CEC2014, and CEC2022) and eleven engineering design problems. In addition, it is compared with fifteen recently proposed algorithms and eleven highly-performing algorithms, such as L-SHADE, LSHADE-cnEpSin, AL-SHADE, mantis search algorithm (MSA), IMODE, AGSK, SOMA_T3A, HyDE-DF, modified LSHADE-SPACMA, SHADE, and LSHADE-SPACMA, to demonstrate its superiority. According to the experimental results, FGO outperforms or is competitive with all of the compared algorithms for the majority of the test functions, implying that it is a high-performing optimizer and a powerful alternative technique for dealing with complex optimization problems. The FGO source code is available on this link</div><div><u>https://drive.mathworks.com/sharing/7b881d79-c7cb-4b64-bdfa-99ab7f57d984</u></div></div>","PeriodicalId":55222,"journal":{"name":"Computer Methods in Applied Mechanics and Engineering","volume":"437 ","pages":"Article 117825"},"PeriodicalIF":6.9,"publicationDate":"2025-02-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143372897","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 : 2025-02-08DOI: 10.1016/j.cma.2025.117806
Santiago Badia , Wei Li , Alberto F. Martín
The use of neural networks to approximate partial differential equations (PDEs) has gained significant attention in recent years. However, the approximation of PDEs with localised phenomena, e.g., sharp gradients and singularities, remains a challenge, due to ill-defined cost functions in terms of pointwise residual sampling or poor numerical integration. In this work, we introduce -adaptive finite element interpolated neural networks. The method relies on the interpolation of a neural network onto a finite element space that is gradually adapted to the solution during the training process to equidistribute a posteriori error indicator. The use of adaptive interpolation is essential in preserving the non-linear approximation capabilities of the neural networks to effectively tackle problems with localised features. The training relies on a gradient-based optimisation of a loss function based on the (dual) norm of the finite element residual of the interpolated neural network. Automatic mesh adaptation (i.e., refinement and coarsening) is performed based on a posteriori error indicators till a certain level of accuracy is reached. The proposed methodology can be applied to indefinite and nonsymmetric problems. We carry out a detailed numerical analysis of the scheme and prove several a priori error estimates, depending on the expressiveness of the neural network compared to the interpolation mesh. Our numerical experiments confirm the effectiveness of the method in capturing sharp gradients and singularities for forward and inverse PDE problems, both in 2D and 3D scenarios. We also show that the proposed preconditioning strategy (i.e., using a dual residual norm of the residual as a cost function) enhances training robustness and accelerates convergence.
{"title":"Adaptive finite element interpolated neural networks","authors":"Santiago Badia , Wei Li , Alberto F. Martín","doi":"10.1016/j.cma.2025.117806","DOIUrl":"10.1016/j.cma.2025.117806","url":null,"abstract":"<div><div>The use of neural networks to approximate partial differential equations (PDEs) has gained significant attention in recent years. However, the approximation of PDEs with localised phenomena, e.g., sharp gradients and singularities, remains a challenge, due to ill-defined cost functions in terms of pointwise residual sampling or poor numerical integration. In this work, we introduce <span><math><mi>h</mi></math></span>-adaptive finite element interpolated neural networks. The method relies on the interpolation of a neural network onto a finite element space that is gradually adapted to the solution during the training process to equidistribute a posteriori error indicator. The use of adaptive interpolation is essential in preserving the non-linear approximation capabilities of the neural networks to effectively tackle problems with localised features. The training relies on a gradient-based optimisation of a loss function based on the (dual) norm of the finite element residual of the interpolated neural network. Automatic mesh adaptation (i.e., refinement and coarsening) is performed based on a posteriori error indicators till a certain level of accuracy is reached. The proposed methodology can be applied to indefinite and nonsymmetric problems. We carry out a detailed numerical analysis of the scheme and prove several a priori error estimates, depending on the expressiveness of the neural network compared to the interpolation mesh. Our numerical experiments confirm the effectiveness of the method in capturing sharp gradients and singularities for forward and inverse PDE problems, both in 2D and 3D scenarios. We also show that the proposed preconditioning strategy (i.e., using a dual residual norm of the residual as a cost function) enhances training robustness and accelerates convergence.</div></div>","PeriodicalId":55222,"journal":{"name":"Computer Methods in Applied Mechanics and Engineering","volume":"437 ","pages":"Article 117806"},"PeriodicalIF":6.9,"publicationDate":"2025-02-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143350928","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 : 2025-02-08DOI: 10.1016/j.cma.2025.117813
Meide Yang, Hongfei Zhang, Dequan Zhang, Fang Wang, Xu Han
Reliability-based design optimization (RBDO) methods based on the most probable point (MPP) have been extensively studied and applied to practical engineering problems. Nevertheless, these methods are not viable when MPP is not straightforward to be searched or multiple MPPs may exhibit. Fortunately, moment method can circumvent the computation of partial derivatives for performance function and iteration to search for MPPs, which is considered as an effective way to solve such problem. However, direct application of moment method to RBDO often incurs high computational cost, which greatly hinders its practicability. To enhance the computational efficiency of the moment-based RBDO methods, an efficient global adaptive Kriging approximation method for RBDO is proposed in this study. The strategy is that a new initial design of experiment scheme according to Gaussian-Hermite integration nodes is innovatively proposed. On this basis, a feasibility check criterion for probabilistic constraints and a selection strategy for candidate samples are respectively proposed to efficiently establish Kriging models of performance functions in the probabilistic constraints. In addition, an enhanced univariate dimension-reduction method with high robustness is presented to calculate the first four-order statistical moments of the above constructed Kriging models. Consequently, the failure probability of each probabilistic constraint can be calculated by Edgeworth series. Finally, a deterministic optimization algorithm is executed to derive the optimal solution. Three numerical examples and two structural examples are exemplified to demonstrate the effectiveness of the proposed moment-based method compared to prevailing MPP-based and Kriging-based RBDO methods.
{"title":"Efficient global adaptive Kriging approximation method in terms of moment for reliability-based design optimization","authors":"Meide Yang, Hongfei Zhang, Dequan Zhang, Fang Wang, Xu Han","doi":"10.1016/j.cma.2025.117813","DOIUrl":"10.1016/j.cma.2025.117813","url":null,"abstract":"<div><div>Reliability-based design optimization (RBDO) methods based on the most probable point (MPP) have been extensively studied and applied to practical engineering problems. Nevertheless, these methods are not viable when MPP is not straightforward to be searched or multiple MPPs may exhibit. Fortunately, moment method can circumvent the computation of partial derivatives for performance function and iteration to search for MPPs, which is considered as an effective way to solve such problem. However, direct application of moment method to RBDO often incurs high computational cost, which greatly hinders its practicability. To enhance the computational efficiency of the moment-based RBDO methods, an efficient global adaptive Kriging approximation method for RBDO is proposed in this study. The strategy is that a new initial design of experiment scheme according to Gaussian-Hermite integration nodes is innovatively proposed. On this basis, a feasibility check criterion for probabilistic constraints and a selection strategy for candidate samples are respectively proposed to efficiently establish Kriging models of performance functions in the probabilistic constraints. In addition, an enhanced univariate dimension-reduction method with high robustness is presented to calculate the first four-order statistical moments of the above constructed Kriging models. Consequently, the failure probability of each probabilistic constraint can be calculated by Edgeworth series. Finally, a deterministic optimization algorithm is executed to derive the optimal solution. Three numerical examples and two structural examples are exemplified to demonstrate the effectiveness of the proposed moment-based method compared to prevailing MPP-based and Kriging-based RBDO methods.</div></div>","PeriodicalId":55222,"journal":{"name":"Computer Methods in Applied Mechanics and Engineering","volume":"437 ","pages":"Article 117813"},"PeriodicalIF":6.9,"publicationDate":"2025-02-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143349293","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 : 2025-02-08DOI: 10.1016/j.cma.2025.117804
Juan M. Gimenez , Francisco M. Sívori , Axel E. Larreteguy , Sabrina I. Montaño , Horacio J. Aguerre , Norberto M. Nigro , Sergio R. Idelsohn
Efficiently simulating turbulent fluid flow within a boundary layer is one of the major challenges in fluid mechanics. While skin friction may have a limited impact on drag at high Reynolds numbers, it plays a crucial role in determining the location of fluid separation points. Shifts in these separation points can dramatically alter drag and lift, underscoring the importance of accurately accounting for viscous effects. It is generally accepted that the Navier–Stokes equations contain all the necessary physical ingredients to accurately simulate fluid flows, even in complex scenarios. With a sufficiently fine mesh, we could simulate all fluid flows without relying on additional empirical approximations. However, this Direct Numerical Simulation (DNS) strategy is computationally impractical with current technology. The Pseudo-DNS (P-DNS) method offers a novel approach to solve the governing equations with the mesh refinement needed to achieve DNS-level accuracy. The solution is divided into fine and coarse scales, and through an iterative process, both scales are solved until convergence. Computational cost is affordable due to parametrize and solving the fine scale under different boundary conditions in simple domains, which allows performing these calculations offline – prior to and independent of the global solution – only once. The key novelty introduced in this work is the wall representative volume element (RVE), which models the time developing of turbulent boundary layers and its outputs can be adapted for adverse and favorable pressure gradient scenarios. The multiscale method enables accurate prediction of aerodynamic forces using relatively coarse meshes for boundary layers, without the need for empirical parameters or case-specific models. Several case studies involving 2D and 3D flows over both streamlined and bluff bodies validate the ability of P-DNS to deliver reliable results while maintaining modest computational requirements.
{"title":"A multiscale Pseudo-DNS approach for solving turbulent boundary-layer problems","authors":"Juan M. Gimenez , Francisco M. Sívori , Axel E. Larreteguy , Sabrina I. Montaño , Horacio J. Aguerre , Norberto M. Nigro , Sergio R. Idelsohn","doi":"10.1016/j.cma.2025.117804","DOIUrl":"10.1016/j.cma.2025.117804","url":null,"abstract":"<div><div>Efficiently simulating turbulent fluid flow within a boundary layer is one of the major challenges in fluid mechanics. While skin friction may have a limited impact on drag at high Reynolds numbers, it plays a crucial role in determining the location of fluid separation points. Shifts in these separation points can dramatically alter drag and lift, underscoring the importance of accurately accounting for viscous effects. It is generally accepted that the Navier–Stokes equations contain all the necessary physical ingredients to accurately simulate fluid flows, even in complex scenarios. With a sufficiently fine mesh, we could simulate all fluid flows without relying on additional empirical approximations. However, this Direct Numerical Simulation (DNS) strategy is computationally impractical with current technology. The Pseudo-DNS (P-DNS) method offers a novel approach to solve the governing equations with the mesh refinement needed to achieve DNS-level accuracy. The solution is divided into fine and coarse scales, and through an iterative process, both scales are solved until convergence. Computational cost is affordable due to parametrize and solving the fine scale under different boundary conditions in simple domains, which allows performing these calculations offline – prior to and independent of the global solution – only once. The key novelty introduced in this work is the wall representative volume element (RVE), which models the time developing of turbulent boundary layers and its outputs can be adapted for adverse and favorable pressure gradient scenarios. The multiscale method enables accurate prediction of aerodynamic forces using relatively coarse meshes for boundary layers, without the need for empirical parameters or case-specific models. Several case studies involving 2D and 3D flows over both streamlined and bluff bodies validate the ability of P-DNS to deliver reliable results while maintaining modest computational requirements.</div></div>","PeriodicalId":55222,"journal":{"name":"Computer Methods in Applied Mechanics and Engineering","volume":"437 ","pages":"Article 117804"},"PeriodicalIF":6.9,"publicationDate":"2025-02-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143372896","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}
扫码关注我们
求助内容：
应助结果提醒方式：
京公网安备 11010802042870号