This study presents an approach to analytical shape sensitivity analysis of linear strain energy in shell structures modeled using thin shell elements. By using the Kirchhoff-Love plate theory and extending conventional finite element methods, the strain energy variations in shell structures due to shape changes are rigorously analyzed in a discrete manner. Sensitivity formulations are sequentially derived by the chain rule, and the procedure to obtain the derivatives of mechanical and geometric properties related to strain energy is explained step by step, ensuring reproducibility and straightforward implementation. Numerical examples include verification of structural analysis results using a benchmark structure and measurement of efficiency and accuracy of our sensitivity analysis implementation compared to the finite difference method. The results with these examples demonstrate the superiority of explicitly computing gradients using the proposed approach, underscoring its potential to advance the optimal design and structural analysis of shell elements.
{"title":"Matrix-Based Shape Sensitivity Analysis for Linear Strain Energy of Triangular Thin Shell Elements","authors":"Kazuki Hayashi, Romain Mesnil","doi":"10.1002/nme.70276","DOIUrl":"https://doi.org/10.1002/nme.70276","url":null,"abstract":"<div>\u0000 \u0000 <p>This study presents an approach to analytical shape sensitivity analysis of linear strain energy in shell structures modeled using thin shell elements. By using the Kirchhoff-Love plate theory and extending conventional finite element methods, the strain energy variations in shell structures due to shape changes are rigorously analyzed in a discrete manner. Sensitivity formulations are sequentially derived by the chain rule, and the procedure to obtain the derivatives of mechanical and geometric properties related to strain energy is explained step by step, ensuring reproducibility and straightforward implementation. Numerical examples include verification of structural analysis results using a benchmark structure and measurement of efficiency and accuracy of our sensitivity analysis implementation compared to the finite difference method. The results with these examples demonstrate the superiority of explicitly computing gradients using the proposed approach, underscoring its potential to advance the optimal design and structural analysis of shell elements.</p>\u0000 </div>","PeriodicalId":13699,"journal":{"name":"International Journal for Numerical Methods in Engineering","volume":"127 3","pages":""},"PeriodicalIF":2.9,"publicationDate":"2026-02-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146680305","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This study presents an approach to analytical shape sensitivity analysis of linear strain energy in shell structures modeled using thin shell elements. By using the Kirchhoff-Love plate theory and extending conventional finite element methods, the strain energy variations in shell structures due to shape changes are rigorously analyzed in a discrete manner. Sensitivity formulations are sequentially derived by the chain rule, and the procedure to obtain the derivatives of mechanical and geometric properties related to strain energy is explained step by step, ensuring reproducibility and straightforward implementation. Numerical examples include verification of structural analysis results using a benchmark structure and measurement of efficiency and accuracy of our sensitivity analysis implementation compared to the finite difference method. The results with these examples demonstrate the superiority of explicitly computing gradients using the proposed approach, underscoring its potential to advance the optimal design and structural analysis of shell elements.
{"title":"Matrix-Based Shape Sensitivity Analysis for Linear Strain Energy of Triangular Thin Shell Elements","authors":"Kazuki Hayashi, Romain Mesnil","doi":"10.1002/nme.70276","DOIUrl":"10.1002/nme.70276","url":null,"abstract":"<div>\u0000 \u0000 <p>This study presents an approach to analytical shape sensitivity analysis of linear strain energy in shell structures modeled using thin shell elements. By using the Kirchhoff-Love plate theory and extending conventional finite element methods, the strain energy variations in shell structures due to shape changes are rigorously analyzed in a discrete manner. Sensitivity formulations are sequentially derived by the chain rule, and the procedure to obtain the derivatives of mechanical and geometric properties related to strain energy is explained step by step, ensuring reproducibility and straightforward implementation. Numerical examples include verification of structural analysis results using a benchmark structure and measurement of efficiency and accuracy of our sensitivity analysis implementation compared to the finite difference method. The results with these examples demonstrate the superiority of explicitly computing gradients using the proposed approach, underscoring its potential to advance the optimal design and structural analysis of shell elements.</p>\u0000 </div>","PeriodicalId":13699,"journal":{"name":"International Journal for Numerical Methods in Engineering","volume":"127 3","pages":""},"PeriodicalIF":2.9,"publicationDate":"2026-02-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146680304","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
A. Vidal-Ferràndiz, A. Carreño, E. Javaloyas, D. Ginestar, G. Verdú
An efficient method for solving large eigenvalue problems efficiently can be developed using hyper-reduced order models, such as those arising from the LU Proper Orthogonal Decomposition (LUPOD). The LUPOD employs dominant orthogonal modes along with a flexible number of collocation points to establish a reduced scalar product, thereby enhancing computational efficiency to construct the reduced order model. This strategy produces accurate results when a similar number of collocation points and orthogonal modes are used. For problems where the number of available modes is small due to the high computational cost of its computation or because sufficient model accuracy can be achieved with less data, LUPOD may not yield enough precise results. This work proposes an extension of the LUPOD method to increase the number of collocation points in the reduced scalar product and consequently, the accuracy of the LUPOD. The performance of this method is illustrated with a diffusion-reaction problem with analytical solution. Then, this method is applied to solve the neutron diffusion eigenvalue problem. Numerical results for these methodologies are tested for obtaining the k-effective and the steady-state neutron flux distribution of three-dimensional nuclear reactor benchmark problems. They show that using 20 snapshots and 40% of the mesh points of the problem as collocation points speeds up the simulations by about 100 times with respect to the resolution of the full order eigenvalue problem while maintaining accurate results that differ by less than 100 pcm in the eigenvalue and less than 1.2% in the neutron flux determination.
{"title":"Hyper-Reduced Model Based on the Proper Orthogonal Decomposition and the LU Factorization Applied to the Neutron Diffusion Eigenvalue Problem","authors":"A. Vidal-Ferràndiz, A. Carreño, E. Javaloyas, D. Ginestar, G. Verdú","doi":"10.1002/nme.70272","DOIUrl":"10.1002/nme.70272","url":null,"abstract":"<p>An efficient method for solving large eigenvalue problems efficiently can be developed using hyper-reduced order models, such as those arising from the LU Proper Orthogonal Decomposition (LUPOD). The LUPOD employs dominant orthogonal modes along with a flexible number of collocation points to establish a reduced scalar product, thereby enhancing computational efficiency to construct the reduced order model. This strategy produces accurate results when a similar number of collocation points and orthogonal modes are used. For problems where the number of available modes is small due to the high computational cost of its computation or because sufficient model accuracy can be achieved with less data, LUPOD may not yield enough precise results. This work proposes an extension of the LUPOD method to increase the number of collocation points in the reduced scalar product and consequently, the accuracy of the LUPOD. The performance of this method is illustrated with a diffusion-reaction problem with analytical solution. Then, this method is applied to solve the neutron diffusion eigenvalue problem. Numerical results for these methodologies are tested for obtaining the k-effective and the steady-state neutron flux distribution of three-dimensional nuclear reactor benchmark problems. They show that using 20 snapshots and 40% of the mesh points of the problem as collocation points speeds up the simulations by about 100 times with respect to the resolution of the full order eigenvalue problem while maintaining accurate results that differ by less than 100 pcm in the eigenvalue and less than 1.2% in the neutron flux determination.</p>","PeriodicalId":13699,"journal":{"name":"International Journal for Numerical Methods in Engineering","volume":"127 3","pages":""},"PeriodicalIF":2.9,"publicationDate":"2026-02-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/nme.70272","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146680303","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
A. Vidal-Ferràndiz, A. Carreño, E. Javaloyas, D. Ginestar, G. Verdú
An efficient method for solving large eigenvalue problems efficiently can be developed using hyper-reduced order models, such as those arising from the LU Proper Orthogonal Decomposition (LUPOD). The LUPOD employs dominant orthogonal modes along with a flexible number of collocation points to establish a reduced scalar product, thereby enhancing computational efficiency to construct the reduced order model. This strategy produces accurate results when a similar number of collocation points and orthogonal modes are used. For problems where the number of available modes is small due to the high computational cost of its computation or because sufficient model accuracy can be achieved with less data, LUPOD may not yield enough precise results. This work proposes an extension of the LUPOD method to increase the number of collocation points in the reduced scalar product and consequently, the accuracy of the LUPOD. The performance of this method is illustrated with a diffusion-reaction problem with analytical solution. Then, this method is applied to solve the neutron diffusion eigenvalue problem. Numerical results for these methodologies are tested for obtaining the k-effective and the steady-state neutron flux distribution of three-dimensional nuclear reactor benchmark problems. They show that using 20 snapshots and 40% of the mesh points of the problem as collocation points speeds up the simulations by about 100 times with respect to the resolution of the full order eigenvalue problem while maintaining accurate results that differ by less than 100 pcm in the eigenvalue and less than 1.2% in the neutron flux determination.
{"title":"Hyper-Reduced Model Based on the Proper Orthogonal Decomposition and the LU Factorization Applied to the Neutron Diffusion Eigenvalue Problem","authors":"A. Vidal-Ferràndiz, A. Carreño, E. Javaloyas, D. Ginestar, G. Verdú","doi":"10.1002/nme.70272","DOIUrl":"https://doi.org/10.1002/nme.70272","url":null,"abstract":"<p>An efficient method for solving large eigenvalue problems efficiently can be developed using hyper-reduced order models, such as those arising from the LU Proper Orthogonal Decomposition (LUPOD). The LUPOD employs dominant orthogonal modes along with a flexible number of collocation points to establish a reduced scalar product, thereby enhancing computational efficiency to construct the reduced order model. This strategy produces accurate results when a similar number of collocation points and orthogonal modes are used. For problems where the number of available modes is small due to the high computational cost of its computation or because sufficient model accuracy can be achieved with less data, LUPOD may not yield enough precise results. This work proposes an extension of the LUPOD method to increase the number of collocation points in the reduced scalar product and consequently, the accuracy of the LUPOD. The performance of this method is illustrated with a diffusion-reaction problem with analytical solution. Then, this method is applied to solve the neutron diffusion eigenvalue problem. Numerical results for these methodologies are tested for obtaining the k-effective and the steady-state neutron flux distribution of three-dimensional nuclear reactor benchmark problems. They show that using 20 snapshots and 40% of the mesh points of the problem as collocation points speeds up the simulations by about 100 times with respect to the resolution of the full order eigenvalue problem while maintaining accurate results that differ by less than 100 pcm in the eigenvalue and less than 1.2% in the neutron flux determination.</p>","PeriodicalId":13699,"journal":{"name":"International Journal for Numerical Methods in Engineering","volume":"127 3","pages":""},"PeriodicalIF":2.9,"publicationDate":"2026-02-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/nme.70272","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146680306","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Philip Cardiff, Dylan Armfield, Željko Tuković, Ivan Batistić
This study proposes a Jacobian-free Newton-Krylov approach for finite-volume solid mechanics. Traditional Newton-based approaches require explicit Jacobian matrix formation and storage, which can be computationally expensive and memory-intensive. In contrast, Jacobian-free Newton-Krylov methods approximate the Jacobian's action using finite differences, combined with Krylov subspace solvers such as the generalised minimal residual method (GMRES), enabling seamless integration into existing segregated finite-volume frameworks without major code refactoring. This work proposes and benchmarks the performance of a compact-stencil Jacobian-free Newton-Krylov method against a conventional segregated approach on a suite of test cases that span varying geometric dimensions, nonlinearities, dynamic responses and material behaviours. Key metrics, including computational cost, memory efficiency and robustness, are evaluated, along with the influence of preconditioning strategies and stabilisation scaling. Results show that the proposed Jacobian-free Newton-Krylov method outperforms the segregated approach in all linear and nonlinear elastic cases, achieving order-of-magnitude speedups in many instances; however, divergence is observed in elastoplastic cases, highlighting areas for further development. It is found that preconditioning choice affects performance: a LU direct solver is fastest for small to moderately sized cases, while a multigrid method is more effective for larger problems. The findings demonstrate that Jacobian-free Newton-Krylov methods are promising for advancing finite-volume solid mechanics simulations, particularly for existing segregated frameworks where minimal modifications enable their adoption. The described implementations are available in the solids4foam toolbox for OpenFOAM, inviting the community to explore, extend and compare these procedures.
{"title":"A Jacobian-Free Newton-Krylov Method for Cell-Centred Finite Volume Solid Mechanics","authors":"Philip Cardiff, Dylan Armfield, Željko Tuković, Ivan Batistić","doi":"10.1002/nme.70268","DOIUrl":"10.1002/nme.70268","url":null,"abstract":"<p>This study proposes a Jacobian-free Newton-Krylov approach for finite-volume solid mechanics. Traditional Newton-based approaches require explicit Jacobian matrix formation and storage, which can be computationally expensive and memory-intensive. In contrast, Jacobian-free Newton-Krylov methods approximate the Jacobian's action using finite differences, combined with Krylov subspace solvers such as the generalised minimal residual method (GMRES), enabling seamless integration into existing segregated finite-volume frameworks without major code refactoring. This work proposes and benchmarks the performance of a compact-stencil Jacobian-free Newton-Krylov method against a conventional segregated approach on a suite of test cases that span varying geometric dimensions, nonlinearities, dynamic responses and material behaviours. Key metrics, including computational cost, memory efficiency and robustness, are evaluated, along with the influence of preconditioning strategies and stabilisation scaling. Results show that the proposed Jacobian-free Newton-Krylov method outperforms the segregated approach in all linear and nonlinear elastic cases, achieving order-of-magnitude speedups in many instances; however, divergence is observed in elastoplastic cases, highlighting areas for further development. It is found that preconditioning choice affects performance: a LU direct solver is fastest for small to moderately sized cases, while a multigrid method is more effective for larger problems. The findings demonstrate that Jacobian-free Newton-Krylov methods are promising for advancing finite-volume solid mechanics simulations, particularly for existing segregated frameworks where minimal modifications enable their adoption. The described implementations are available in the solids4foam toolbox for OpenFOAM, inviting the community to explore, extend and compare these procedures.</p>","PeriodicalId":13699,"journal":{"name":"International Journal for Numerical Methods in Engineering","volume":"127 3","pages":""},"PeriodicalIF":2.9,"publicationDate":"2026-02-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/nme.70268","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146130012","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Muhammad Babar Shamim, Hauke Goldbeck, Stephan Wulfinghoff
This work presents a fully thermomechanically coupled material model for shape memory alloys (SMAs), capable of predicting shape memory effect, superelasticity, stress and strain recovery, and martensite reorientation. Formulated within the Generalized Standard Material (GSM) framework, the model employs a rate potential, whose variations yield the governing equations, including linear momentum balance, energy balance, and evolution of internal variables. A potential-based line search method integrated with a Newton–Raphson scheme enhances the robustness and convergence of the solution algorithm. Extending the Sedlák [14] model's energy and dissipation formulations, we apply the proposed framework to an SMA-based out-of-plane bistable microactuator design. The actuator features two antagonistically coupled SMA microbridges and exhibits bistable behavior, snapping between stable states under thermomechanical loading and using constrained recovery forces to perform work. Results demonstrate the model's efficiency and accuracy in capturing the complex thermomechanical response of SMA devices, highlighting its potential for advanced bistable actuator design.
{"title":"Variational Thermomechanically Coupled Shape Memory Alloy Material Model and Optimization of Shape Memory Alloy Based Out-of-Plane Bistable Microactuator","authors":"Muhammad Babar Shamim, Hauke Goldbeck, Stephan Wulfinghoff","doi":"10.1002/nme.70263","DOIUrl":"10.1002/nme.70263","url":null,"abstract":"<p>This work presents a fully thermomechanically coupled material model for shape memory alloys (SMAs), capable of predicting shape memory effect, superelasticity, stress and strain recovery, and martensite reorientation. Formulated within the Generalized Standard Material (GSM) framework, the model employs a rate potential, whose variations yield the governing equations, including linear momentum balance, energy balance, and evolution of internal variables. A potential-based line search method integrated with a Newton–Raphson scheme enhances the robustness and convergence of the solution algorithm. Extending the Sedlák [14] model's energy and dissipation formulations, we apply the proposed framework to an SMA-based out-of-plane bistable microactuator design. The actuator features two antagonistically coupled SMA microbridges and exhibits bistable behavior, snapping between stable states under thermomechanical loading and using constrained recovery forces to perform work. Results demonstrate the model's efficiency and accuracy in capturing the complex thermomechanical response of SMA devices, highlighting its potential for advanced bistable actuator design.</p>","PeriodicalId":13699,"journal":{"name":"International Journal for Numerical Methods in Engineering","volume":"127 3","pages":""},"PeriodicalIF":2.9,"publicationDate":"2026-02-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/nme.70263","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146130014","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The symplectic algorithm for Hamiltonian systems is renowned for its high accuracy and efficiency in long-term computations. However, its direct application to non-conservative problems in engineering dynamics is limited by the presence of dissipative effects and external load terms. To overcome these challenges, this paper presents a novel symplectic precise integration method based on the perturbation series approach, specifically tailored for structural dynamic response problems. By introducing a state vector, the general structural dynamic response equation is reformulated into a generalized Hamiltonian framework. Subsequently, perturbation theory is applied to transform this generalized Hamiltonian equation into a sequence of linear Hamiltonian equations, effectively converting the problem of solving non-conservative equations into one of solving multiple conservative equations. The precise integration method is then used to derive a step mapping matrix that significantly improves accuracy without substantially increasing computational cost. Numerical examples are provided to highlight the efficacy and practical applicability of the proposed method in addressing complex structural dynamic response problems.
{"title":"A Symplectic Precise Integration Perturbation Series Method for Linear Structural Dynamic Response Problems","authors":"Zhiping Qiu, Yu Qiu","doi":"10.1002/nme.70271","DOIUrl":"10.1002/nme.70271","url":null,"abstract":"<div>\u0000 \u0000 <p>The symplectic algorithm for Hamiltonian systems is renowned for its high accuracy and efficiency in long-term computations. However, its direct application to non-conservative problems in engineering dynamics is limited by the presence of dissipative effects and external load terms. To overcome these challenges, this paper presents a novel symplectic precise integration method based on the perturbation series approach, specifically tailored for structural dynamic response problems. By introducing a state vector, the general structural dynamic response equation is reformulated into a generalized Hamiltonian framework. Subsequently, perturbation theory is applied to transform this generalized Hamiltonian equation into a sequence of linear Hamiltonian equations, effectively converting the problem of solving non-conservative equations into one of solving multiple conservative equations. The precise integration method is then used to derive a step mapping matrix that significantly improves accuracy without substantially increasing computational cost. Numerical examples are provided to highlight the efficacy and practical applicability of the proposed method in addressing complex structural dynamic response problems.</p>\u0000 </div>","PeriodicalId":13699,"journal":{"name":"International Journal for Numerical Methods in Engineering","volume":"127 3","pages":""},"PeriodicalIF":2.9,"publicationDate":"2026-02-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146130013","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The uncertainty in a vehicle's position and dynamic state at the end of the Mars atmospheric entry significantly influences the successful deployment of the parachute. This uncertainty is primarily driven by measurement uncertainties, including those associated with the vehicle's initial state and model parameters. These uncertainties propagate through the dynamic system, thereby contributing to the overall uncertainty in the vehicle's terminal state. Traditionally, errors in the vehicle's position and dynamic state are evaluated separately using different indicators. However, successful parachute deployment depends on both the vehicle's position and dynamic state. This paper presents a method that integrates errors in both the vehicle's position and dynamic state to quantify the uncertainty in the terminal state of Mars entry vehicles, considering various contributing factors. The method also assesses the probability of successful parachute deployment. First, a covariance matrix representing the vehicle's terminal state error is derived by propagating measurement uncertainties through the dynamic system using linear covariance analysis. A terminal state ellipsoid model is developed. Subsequently, an altitude-velocity-downrange parachute feasible deployment box is defined, considering the constraints in altitude, Mach number, dynamic pressure, and deployment position accuracy. By integrating the terminal state ellipsoid with the feasible deployment box, an integral probability model is constructed to estimate the maximum probability of successful parachute deployment within the predefined box. Numerical simulations validated the effectiveness of the terminal state ellipsoid and the integral probability model in quantifying uncertainty and demonstrating their capability to assess the vehicle's terminal position and to ensure deployment safety.
{"title":"Ellipsoid-Based Integral Probability Model for Vehicle Position and Dynamic State Evaluation During Parachute Deployment in Mars EDL Missions","authors":"Yanmin Jin, Zhixian Luo, Xiaohua Tong, Huan Xie, Changjiang Xiao, Yongjiu Feng, Xiong Xu","doi":"10.1002/nme.70270","DOIUrl":"https://doi.org/10.1002/nme.70270","url":null,"abstract":"<div>\u0000 \u0000 <p>The uncertainty in a vehicle's position and dynamic state at the end of the Mars atmospheric entry significantly influences the successful deployment of the parachute. This uncertainty is primarily driven by measurement uncertainties, including those associated with the vehicle's initial state and model parameters. These uncertainties propagate through the dynamic system, thereby contributing to the overall uncertainty in the vehicle's terminal state. Traditionally, errors in the vehicle's position and dynamic state are evaluated separately using different indicators. However, successful parachute deployment depends on both the vehicle's position and dynamic state. This paper presents a method that integrates errors in both the vehicle's position and dynamic state to quantify the uncertainty in the terminal state of Mars entry vehicles, considering various contributing factors. The method also assesses the probability of successful parachute deployment. First, a covariance matrix representing the vehicle's terminal state error is derived by propagating measurement uncertainties through the dynamic system using linear covariance analysis. A terminal state ellipsoid model is developed. Subsequently, an altitude-velocity-downrange parachute feasible deployment box is defined, considering the constraints in altitude, Mach number, dynamic pressure, and deployment position accuracy. By integrating the terminal state ellipsoid with the feasible deployment box, an integral probability model is constructed to estimate the maximum probability of successful parachute deployment within the predefined box. Numerical simulations validated the effectiveness of the terminal state ellipsoid and the integral probability model in quantifying uncertainty and demonstrating their capability to assess the vehicle's terminal position and to ensure deployment safety.</p>\u0000 </div>","PeriodicalId":13699,"journal":{"name":"International Journal for Numerical Methods in Engineering","volume":"127 3","pages":""},"PeriodicalIF":2.9,"publicationDate":"2026-02-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146136035","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The uncertainty in a vehicle's position and dynamic state at the end of the Mars atmospheric entry significantly influences the successful deployment of the parachute. This uncertainty is primarily driven by measurement uncertainties, including those associated with the vehicle's initial state and model parameters. These uncertainties propagate through the dynamic system, thereby contributing to the overall uncertainty in the vehicle's terminal state. Traditionally, errors in the vehicle's position and dynamic state are evaluated separately using different indicators. However, successful parachute deployment depends on both the vehicle's position and dynamic state. This paper presents a method that integrates errors in both the vehicle's position and dynamic state to quantify the uncertainty in the terminal state of Mars entry vehicles, considering various contributing factors. The method also assesses the probability of successful parachute deployment. First, a covariance matrix representing the vehicle's terminal state error is derived by propagating measurement uncertainties through the dynamic system using linear covariance analysis. A terminal state ellipsoid model is developed. Subsequently, an altitude-velocity-downrange parachute feasible deployment box is defined, considering the constraints in altitude, Mach number, dynamic pressure, and deployment position accuracy. By integrating the terminal state ellipsoid with the feasible deployment box, an integral probability model is constructed to estimate the maximum probability of successful parachute deployment within the predefined box. Numerical simulations validated the effectiveness of the terminal state ellipsoid and the integral probability model in quantifying uncertainty and demonstrating their capability to assess the vehicle's terminal position and to ensure deployment safety.
{"title":"Ellipsoid-Based Integral Probability Model for Vehicle Position and Dynamic State Evaluation During Parachute Deployment in Mars EDL Missions","authors":"Yanmin Jin, Zhixian Luo, Xiaohua Tong, Huan Xie, Changjiang Xiao, Yongjiu Feng, Xiong Xu","doi":"10.1002/nme.70270","DOIUrl":"10.1002/nme.70270","url":null,"abstract":"<div>\u0000 \u0000 <p>The uncertainty in a vehicle's position and dynamic state at the end of the Mars atmospheric entry significantly influences the successful deployment of the parachute. This uncertainty is primarily driven by measurement uncertainties, including those associated with the vehicle's initial state and model parameters. These uncertainties propagate through the dynamic system, thereby contributing to the overall uncertainty in the vehicle's terminal state. Traditionally, errors in the vehicle's position and dynamic state are evaluated separately using different indicators. However, successful parachute deployment depends on both the vehicle's position and dynamic state. This paper presents a method that integrates errors in both the vehicle's position and dynamic state to quantify the uncertainty in the terminal state of Mars entry vehicles, considering various contributing factors. The method also assesses the probability of successful parachute deployment. First, a covariance matrix representing the vehicle's terminal state error is derived by propagating measurement uncertainties through the dynamic system using linear covariance analysis. A terminal state ellipsoid model is developed. Subsequently, an altitude-velocity-downrange parachute feasible deployment box is defined, considering the constraints in altitude, Mach number, dynamic pressure, and deployment position accuracy. By integrating the terminal state ellipsoid with the feasible deployment box, an integral probability model is constructed to estimate the maximum probability of successful parachute deployment within the predefined box. Numerical simulations validated the effectiveness of the terminal state ellipsoid and the integral probability model in quantifying uncertainty and demonstrating their capability to assess the vehicle's terminal position and to ensure deployment safety.</p>\u0000 </div>","PeriodicalId":13699,"journal":{"name":"International Journal for Numerical Methods in Engineering","volume":"127 3","pages":""},"PeriodicalIF":2.9,"publicationDate":"2026-02-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146135925","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Encoding velocity fields as quantum states poses a significant challenge in the development of quantum algorithms for fluid dynamics. Conventional methods, often based on direct normalization of discrete velocity data, do not intrinsically capture the structure and dynamics of vortex flows, potentially introducing artifacts that affect accuracy in modeling flow evolution. We propose a method for encoding velocity fields as quantum states of a spinor field using the spherical Clebsch representation. By applying a pointwise normalization constraint, we develop an ansatz with parameterized controlled rotation gates, optimized through a variational quantum algorithm. This approach encodes target velocity fields into spinor-based quantum states, offering a pathway to more efficient quantum simulations of fluid dynamics. Furthermore, the encoded quantum state can be mapped to the vortex-surface field, providing a useful approach for analyzing vortex dynamics and characterizing flow structures. While the calculation of the loss function encounters exponential complexity per training step, and the measurement of all qubits is inevitable, leading to high computational complexity for implementation on quantum hardware and requiring further optimization, its effectiveness has been validated across various scenarios through quantum simulation. This method enables spinor-based encoding and quantum simulation, with potential applications to diverse vector fields and complex flows, including magnetohydrodynamics and reactive flows.
{"title":"Quantum State Encoding of Vortical Flows With the Spinor Field","authors":"Hao Su, Shiying Xiong, Yue Yang","doi":"10.1002/nme.70267","DOIUrl":"10.1002/nme.70267","url":null,"abstract":"<div>\u0000 \u0000 <p>Encoding velocity fields as quantum states poses a significant challenge in the development of quantum algorithms for fluid dynamics. Conventional methods, often based on direct normalization of discrete velocity data, do not intrinsically capture the structure and dynamics of vortex flows, potentially introducing artifacts that affect accuracy in modeling flow evolution. We propose a method for encoding velocity fields as quantum states of a spinor field using the spherical Clebsch representation. By applying a pointwise normalization constraint, we develop an ansatz with parameterized controlled rotation gates, optimized through a variational quantum algorithm. This approach encodes target velocity fields into spinor-based quantum states, offering a pathway to more efficient quantum simulations of fluid dynamics. Furthermore, the encoded quantum state can be mapped to the vortex-surface field, providing a useful approach for analyzing vortex dynamics and characterizing flow structures. While the calculation of the loss function encounters exponential complexity per training step, and the measurement of all qubits is inevitable, leading to high computational complexity for implementation on quantum hardware and requiring further optimization, its effectiveness has been validated across various scenarios through quantum simulation. This method enables spinor-based encoding and quantum simulation, with potential applications to diverse vector fields and complex flows, including magnetohydrodynamics and reactive flows.</p>\u0000 </div>","PeriodicalId":13699,"journal":{"name":"International Journal for Numerical Methods in Engineering","volume":"127 3","pages":""},"PeriodicalIF":2.9,"publicationDate":"2026-02-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146129949","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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