International Journal for Numerical Methods in Engineering最新文献

{"title":"A Stable Poro-Mechanical Formulation for Material Point Methods Leveraging Overlapping Meshes and Multi-Field Ghost Penalisation","authors":"Giuliano Pretti,&nbsp;Robert E. Bird,&nbsp;Nathan D. Gavin,&nbsp;William M. Coombs,&nbsp;Charles E. Augarde","doi":"10.1002/nme.7630","DOIUrl":"https://doi.org/10.1002/nme.7630","url":null,"abstract":"<p>The Material Point Method (MPM) is widely used to analyse coupled (solid-water) problems under large deformations/displacements. However, if not addressed carefully, MPM u-p formulations for poromechanics can be affected by two major sources of instability. Firstly, inf-sup condition violation can arise when the spaces for the displacement and pressure fields are not chosen correctly, resulting in an unstable pressure field when the equations are monolithically solved. Secondly, the intrinsic nature of particle-based discretisation makes the MPM an unfitted mesh-based method, which can affect the system's condition number and solvability, particularly when background mesh elements are poorly populated. This work proposes a solution to both problems. The inf-sup condition is avoided using two overlapping meshes, a coarser one for the pressure and a finer one for the displacement. This approach does not require stabilisation of the primary equations since it is stable by design and is particularly valuable for low-order shape functions. As for the system's poor condition number, a face ghost penalisation method is added to both the primary equations, which constitutes a novelty in the context of MPM mixed formulations. This study frequently makes use of the theories of functional analysis or the unfitted Finite Element Method (FEM). Although these theories may not directly apply to the MPM, they provide a robust and logical basis for the research. These rationales are further supported by four numerical examples, which encompass both elastic and elasto-plastic cases and drained and undrained conditions.</p>","PeriodicalId":13699,"journal":{"name":"International Journal for Numerical Methods in Engineering","volume":"126 5","pages":""},"PeriodicalIF":2.7,"publicationDate":"2025-03-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/nme.7630","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143533955","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}

{"title":"Multi-Phase-Field Method for Dynamic Fracture in Composite Materials Based on Reduced-Order-Homogenization","authors":"Nianqi Liu,&nbsp;Zifeng Yuan","doi":"10.1002/nme.70012","DOIUrl":"https://doi.org/10.1002/nme.70012","url":null,"abstract":"<div>\u0000 \u0000 <p>In this manuscript, we extend the quasi-static multi-phase-field method for composite materials to the dynamic case. In the dynamic multi-phase-field method, each phase of the composites has its individual phase field, and the degradation of each phase is governed by its respective phase field. The macroscopic response is then obtained by averaging and homogenization approaches through the reduced-order-homogenization (ROH) framework. Through the ROH and the <span>Francfort-Marigo</span> variational principle, we can obtain the equations that govern the motion of the composites and the evolution of each phase field. This method is capable of capturing the characteristics of dynamic fracture, such as crack branching, without the need for any additional bifurcation criterion. Moreover, it can capture dynamic fracture patterns in composite materials, including matrix cracking, fiber breakage, and delamination. The corresponding numerical algorithm that includes spatial and temporal discretization is developed. An implicit, staggered <span>Newton-Raphson</span> iterative scheme is implemented to solve the nonlinear coupled equations. Finally, this method is tested with several sets of dynamic fracture benchmarks, which demonstrates good agreement with the experiments and other numerical methods.</p>\u0000 </div>","PeriodicalId":13699,"journal":{"name":"International Journal for Numerical Methods in Engineering","volume":"126 4","pages":""},"PeriodicalIF":2.7,"publicationDate":"2025-02-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143481406","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}

{"title":"Multiscale Topology Optimization Applying FFT-Based Homogenization","authors":"Masayoshi Matsui,&nbsp;Hiroya Hoshiba,&nbsp;Koji Nishiguchi,&nbsp;Hiroki Ogura,&nbsp;Junji Kato","doi":"10.1002/nme.70009","DOIUrl":"https://doi.org/10.1002/nme.70009","url":null,"abstract":"<div>\u0000 \u0000 <p>Advances in 3D-printing technology have enabled the fabrication of periodic microstructures that exhibit characteristic mechanical performances. In response, multiscale topology optimization, which finds the optimal design of microstructure for the macrostructure geometry and performance requirements, has become a hot topic in the field of structural optimization. While the basic optimization framework based on the homogenization theory spanning macro and microscales is available, it is computationally expensive and not easily applicable in practical scenarios such as high-resolution design for precision modeling and reliable design considering non-linearities. To address this issue, we focus on a homogenization analysis using a fast Fourier transform as an alternative approach to conventional finite element analysis and develop an optimization method with fast computing speed and low memory requirements. In this paper, we define a simple stiffness maximization problem with linear elastic materials and conduct two and three-dimensional optimization analyses to evaluate the validity and performance of the proposed method. We discuss the advantages of computational cost, the influence of the filtering process, and the appropriate setting of material contrast.</p>\u0000 </div>","PeriodicalId":13699,"journal":{"name":"International Journal for Numerical Methods in Engineering","volume":"126 4","pages":""},"PeriodicalIF":2.7,"publicationDate":"2025-02-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143447026","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}

{"title":"Construction of a \u0000 \u0000 \u0000 \u0000 \u0000 C\u0000 \u0000 \u0000 1\u0000 \u0000 \u0000 \u0000 $$ {C}^1 $$\u0000 Polygonal Spline Element Based on the Scaled Boundary Coordinates","authors":"Zhen-Yi Liu,&nbsp;Chong-Jun Li","doi":"10.1002/nme.7671","DOIUrl":"https://doi.org/10.1002/nme.7671","url":null,"abstract":"&lt;div&gt;\u0000 \u0000 &lt;p&gt;We construct a new polygonal &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;msup&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;C&lt;/mi&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mn&gt;1&lt;/mn&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/msup&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt;$$ {C}^1 $$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; spline finite element method based on the scaled boundary coordinates to address the plate bending problems in the Kirchhoff-love formulation. The Bernstein interpolations are utilized in both radial and circumferential directions in the scaled boundary coordinates. Firstly, the &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;msup&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;C&lt;/mi&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mn&gt;1&lt;/mn&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/msup&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt;$$ {C}^1 $$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; continuity conditions inside an S-domain and normal derivatives constraining conditions are imposed by a simple linear system on the S-net coefficients. Secondly, to satisfy the &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;msup&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;C&lt;/mi&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mn&gt;1&lt;/mn&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/msup&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt;$$ {C}^1 $$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; connection between different polygonal elements, we construct the Hermite interpolation by equivalently transforming part of the S-net coefficients to proper boundary degrees of freedom, namely, three degrees of freedom at each vertex and a normal derivative at the midpoint of each edge. Moreover, we discuss the convergence analysis of the proposed element over convex meshes by finding the necessary and sufficient geometric conditions, where the corresponding unisolvency theorem is proved by studying the dimension of the spline space &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;mrow&gt;\u0000 &lt;msubsup&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;S&lt;/mi&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mn&gt;4&lt;/mn&gt;\u0000 &lt;mo&gt;,&lt;/mo&gt;\u0000 &lt;mn&gt;3&lt;/mn&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mn&gt;1&lt;/mn&gt;\u0000 &lt;mo&gt;,&lt;/mo&gt;\u0000 &lt;mo&gt;∗&lt;/mo&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/msubsup&gt;\u0000 &lt;mo&gt;(&lt;/mo&gt;\u0000 &lt;msub&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;𝒯&lt;/mi&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;S&lt;/mi&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/msub&gt;\u0000 &lt;","PeriodicalId":13699,"journal":{"name":"International Journal for Numerical Methods in Engineering","volume":"126 4","pages":""},"PeriodicalIF":2.7,"publicationDate":"2025-02-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143447203","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}

{"title":"A Novel Neural Network Model and Two New Algorithms for Solving Multiobjective Linear Optimization Problems","authors":"Mahboobe Abkhizi,&nbsp;Mehrdad Ghaznavi,&nbsp;Mohammad Hadi Noori Skandari","doi":"10.1002/nme.7670","DOIUrl":"https://doi.org/10.1002/nme.7670","url":null,"abstract":"<div>\u0000 \u0000 <p>In this article, the Pareto front of multiobjective linear optimization problems (MLOPs) is approximated via a new neural network (NN) model. Karush-Kuhn-Tucker (KKT) optimality conditions for multiobjective linear optimization problems are applied to construct this neural network model. Compared with the available models in the literature, the proposed approach employs the KKT optimality conditions of the main MLOP, not a scalarized problem related to the MLOP. The stability of the suggested NN model in the sense of Lyapunov, is proved. Also, it is shown that the proposed NN is globally convergent to an efficient solution of the main MLOP. Moreover, we present two algorithms to attain some nondominated points with equidistant distribution throughout the Pareto front of bi-objective and three-objective optimization problems. In the suggested algorithms we apply some filters to attain a uniform approximation of the Pareto front. Illustrative results are provided to clarify the validity and performance of the introduced model for different categories of MLOPs. Numerical results satisfy the presented theoretical aspects. In order to have a comparison with other methods, three indicators, including Hypervolume (HV), Spacing, and Even distribution (EV), are utilized. Finally, we apply the proposed idea for the sustainable development of a multinational company in automotive engineering.</p>\u0000 </div>","PeriodicalId":13699,"journal":{"name":"International Journal for Numerical Methods in Engineering","volume":"126 4","pages":""},"PeriodicalIF":2.7,"publicationDate":"2025-02-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143447030","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}

{"title":"New Semi-Analytical Shell Elements","authors":"Jianghuai Li","doi":"10.1002/nme.70011","DOIUrl":"https://doi.org/10.1002/nme.70011","url":null,"abstract":"<div>\u0000 \u0000 <p>New semi-analytical shell elements are developed using the scaled boundary finite element method. A shell element is treated as a three-dimensional continuum whose midsurface, the generalized “boundary” of the continuum, is characterized with a quadrilateral spectral element. It is along the thickness direction <i>ξ</i> that the midsurface is scaled to represent the three-dimensional geometry and that the analytical displacement solution is sought. Neumann expansion is used to approximate the inverse of the Jacobian as a quadratic matrix polynomial of <i>ξ</i> while the assumed natural strain method is applied to alleviate the shear and membrane locking. The virtual work principle considering body forces is utilized to derive the scaled boundary finite element equation, which is directly solved via the differential quadrature method. Numerical examples show that the shell elements with five displacement sampling points along <i>ξ</i> can efficiently analyze thin to very thick general shells.</p>\u0000 </div>","PeriodicalId":13699,"journal":{"name":"International Journal for Numerical Methods in Engineering","volume":"126 4","pages":""},"PeriodicalIF":2.7,"publicationDate":"2025-02-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143456118","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}

{"title":"Construction of Data Sequence for Model Order Reduction in Thermomechanical Modeling of DED Additive Manufacturing","authors":"Joël Keumo Tematio,&nbsp;David Ryckelynck,&nbsp;Michel Bellet,&nbsp;Yancheng Zhang","doi":"10.1002/nme.70005","DOIUrl":"https://doi.org/10.1002/nme.70005","url":null,"abstract":"<div>\u0000 \u0000 <p>Reduced order modeling (ROM) is applied to the finite element thermo-mechanical simulation of metal additive manufacturing at part scale. This is a significant challenge because of the continuously evolving computational domain, on which a local reduced basis is required to apply the projection-based ROM. In this paper, ROM is applied to the mechanical resolution, which is much more time-consuming than the thermal one. Considering the modeling of DED processes (directed energy deposition), it is proposed to organize the training set of simulation snapshots according to an energy deposition length that represents the progress of the process. The full-order model consists of a transient thermomechanical model modified by use of the previously developed Inherent Strain Rate method. When applying the projection-based ROM to this full-order model, the constructed data sequence enables the design a local ROM depending on the energy deposition length and process parameters. The approach, in its present state, is limited to constructions with a constant transverse geometry and a constant set of process parameters. The simulation of the DED construction of a turbine blade mock-up, made of thirty layers with interlayer dwell times, revealed a computational speedup of about 100.</p>\u0000 </div>","PeriodicalId":13699,"journal":{"name":"International Journal for Numerical Methods in Engineering","volume":"126 4","pages":""},"PeriodicalIF":2.7,"publicationDate":"2025-02-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143447025","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}

{"title":"Data-Based Estimation of Critical Time Steps for Explicit Time Integration","authors":"Tobias Willmann,&nbsp;Maximilian Schilling,&nbsp;Manfred Bischoff","doi":"10.1002/nme.7666","DOIUrl":"https://doi.org/10.1002/nme.7666","url":null,"abstract":"<p>Finding the critical time step for conditionally stable time integration methods has been a decades-long problem. The apparently obvious option of directly computing it from a generalized eigenvalue analysis, identifying the largest eigenfrequency of the discrete system, is usually impractical because of its numerical expense and because a stiffness matrix is often unavailable in the context of explicit analysis. There exist two popular approaches to efficiently estimate the critical time step: A characteristic element length can be estimated based on heuristic formulas. The resulting estimate, however, cannot be guaranteed to be conservative. Another approach is to reformulate and simplify the underlying eigenvalue problem on the element level and to use certain inequalities to derive an upper bound for the largest eigenvalue. This is conservative but may show poor performance by significantly under-predicting the actual critical time step. Moreover, the necessary simplifications are usually specific to the investigated element formulation. Many works that develop time step estimators demonstrate their performance only for particular element configurations, making it difficult to compare the estimators. In this paper, data-driven approaches for time step estimation for 2d-elements that address several of the aforementioned problems are proposed. First, the set of all possible quadrilateral element geometries and its discrete representation are described. A detailed comparison of nine existing time step estimators based on more than ten million element configurations is presented. Additionally, the concept of an optimal safety factor function is introduced. This concept allows us to generate the optimal and conservative version of an existing estimator and thus solves two problems at the same time: It can be used to make non-conservative estimators conservative and to improve the performance of estimators that are conservative by construction. Finally, we formulate time step estimation as a function approximation problem. It allows us to derive customizable time step estimators solely based on data. Through two examples, we demonstrate that this data-driven approach yields time step estimators that outperform state-of-the-art estimators in terms of accuracy while also being efficient to evaluate.</p>","PeriodicalId":13699,"journal":{"name":"International Journal for Numerical Methods in Engineering","volume":"126 4","pages":""},"PeriodicalIF":2.7,"publicationDate":"2025-02-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/nme.7666","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143447031","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}

{"title":"A Petrov-Galerkin Dual-Porosity Framework for Thermal Analysis of Fractured Porous Media","authors":"Mahtab Taghvaei,&nbsp;Amir R. Khoei","doi":"10.1002/nme.7656","DOIUrl":"https://doi.org/10.1002/nme.7656","url":null,"abstract":"<div>\u0000 \u0000 <p>Thermal analysis of fractured porous media is of interest in environmental and reservoir engineering. The dual-porosity method is a common and cost-efficient approach for modeling fractured domain. In the case of thermal analysis, this method encounters two challenges; the high dependence on the matrix to fracture transfer parameter, “shape factor,” and numerical oscillations produced in the conventional Galerkin finite element formulation for convection dominant problems. To overcome these issues a Petrov-Galerkin dual porosity (PGDP) algorithm is presented for the analysis of 2D transient heat flow in fractured porous media. In the computational algorithm, an appropriate unit cell is selected from the primary domain and the corresponding thermal shape factor is evaluated with a time-varying function. Unlike conventional constant shape factor models, this shape factor accounts for not only the geometry of the blocks, but also the thermal properties of the domain and existing boundary conditions. Several case studies are simulated to investigate the validity and sensitivity of the time-dependent thermal shape factor to different parameters for square and non-square blocks. Moreover, the Petrov-Galerkin formulation is applied to effectively achieve the accurate spatial results for highly convective heat flows. Numerical simulations are performed to study the efficiency and accuracy of the proposed PGDP algorithm for different range of convection and conduction regimes. This study illustrates that the PGDP algorithm efficiently enhances the accuracy of the simulation in modeling of fractured domains, particularly in transient stages. Moreover, the capability of the proposed computational algorithm is demonstrated in modeling square and non-square matrix formations.</p>\u0000 </div>","PeriodicalId":13699,"journal":{"name":"International Journal for Numerical Methods in Engineering","volume":"126 3","pages":""},"PeriodicalIF":2.7,"publicationDate":"2025-02-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143362632","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}

{"title":"The Failure Prediction of Reinforced Composite Quasi-Brittle Structures by an Improved Version of the Extended Lumped Damage Approach","authors":"Daniel V. C. Teles,&nbsp;David L. N. F. Amorim,&nbsp;Edson D. Leonel","doi":"10.1002/nme.70006","DOIUrl":"https://doi.org/10.1002/nme.70006","url":null,"abstract":"<div>\u0000 \u0000 <p>This study presents an improved version of the Extended Lumped Damage Mechanics (XLDM) formulation within a position-based approach of the Finite Element Method (FEM). In the XLDM, the strain field has been assessed from the elongations of numerical extensometers, which connect the finite element nodes. In addition, localisation bands positioned along the elements' boundaries depict the mechanical effects of material degradation. The position-based approach of FEM enables the accurate modelling of geometrically non-linear effects and its computational implementation is straightforward. In this approach, the equilibrium configuration has been evaluated in relation to the nodal positions instead of its displacements. Thus, one improvement proposed herein involves the coupling of the XLDM failure predictions within an exact geometrically non-linear framework. Besides, in this study, the XLDM has been further improved by incorporating the damage growth caused by compressive stresses. The non-linear formulation proposed herein enables the presence of reinforcements, which have been added by an embedded scheme and lead to another improvement in the XLDM context. Three applications demonstrate the accuracy of the proposed non-linear scheme, in which the numerical responses obtained by the proposed improved formulation have been compared to experimental results available in the literature.</p>\u0000 </div>","PeriodicalId":13699,"journal":{"name":"International Journal for Numerical Methods in Engineering","volume":"126 3","pages":""},"PeriodicalIF":2.7,"publicationDate":"2025-02-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143362634","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}