International Journal for Numerical Methods in Engineering最新文献

Model reduction is an active research field to construct low-dimensional surrogate models of high fidelity to accelerate engineering design cycles. In this work, we investigate model reduction for linear structured systems using dominant reachable and observable subspaces. When the training set—containing all possible interpolation points—is large, these subspaces can be determined by solving many large-scale linear systems. However, for high-dimensional models, this easily becomes computationally intractable. To circumvent this issue, in this work, we propose an active sampling strategy to sample only a few points from the given training set, which can allow us to estimate those subspaces accurately. To this end, we formulate the identification of the subspaces as the solution of the generalized Sylvester equations, guiding us to select the most relevant samples from the training set to achieve our goals. Consequently, we construct solutions of the matrix equations in low-rank forms, which encode subspace information. We extensively discuss computational aspects and efficient usage of the low-rank factors in the process of obtaining reduced-order models. We illustrate the proposed active sampling scheme to obtain reduced-order models via dominant reachable and observable subspaces and present its comparison with the method where all the points from the training set are taken into account. It is shown that the active sample strategy can provide us a speed-up by one order of magnitude without sacrificing any noticeable accuracy.

Parametric model order reduction by matrix interpolation allows for efficient prediction of the behavior of dynamic systems without requiring knowledge about the underlying parametric dependency. Within this approach, reduced models are first sampled and then made consistent with each other by transforming the underlying reduced bases. Finally, the transformed reduced operators can be interpolated to predict reduced models for queried parameter points. However, the accuracy of the predicted reduced model strongly depends on the similarity of the sampled reduced bases. If the local reduced bases change significantly over the parameter space, inconsistencies are introduced in the training data for the matrix interpolation. These strong changes in the reduced bases can occur due to the model order reduction method used, a change of the system's dynamics with a change of the parameters, and mode switching and truncation. In this paper, individual approaches for removing these inconsistencies are extended and combined into one general framework to simultaneously treat multiple sources of inconsistency. For that, modal truncation is used for the reduction, an adaptive sampling of the parameter space is performed, and eventually, the parameter space is partitioned into regions in which all local reduced bases are consistent with those of their neighboring samples within the same region. The proposed framework is applied to a cantilever Timoshenko beam and the Kelvin cell for one- to three-dimensional parameter spaces. Compared to the original version of parametric model order reduction by matrix interpolation and an existing method for inconsistency removal, the proposed framework leads to parametric reduced models with significantly smaller errors.

This paper introduces an unprecedented unified approach for developing structural theories with an arbitrary kinematic variable over the beam cross-section. Each of the three displacement variables can be analyzed using an independent expansion function. Both the order of the expansion and the number of terms in each field can be any. That is, the same order does not necessarily correspond to the same number of unknown variables. This method permits starting from a general model and write classical and known higher-order beam theories without any restrain. In this paper, the structural theories are built by using the polynomial expansion of the cross-sectional variables. The Carrera unified formulation (CUF) is employed to describe the cross-sectional kinematics. The finite element method (FEM) is employed to discretize the structure along the beam axis, utilizing Lagrange-based elements. The governing equations and related FE arrays for linear analysis are derived using the principle of virtual displacements. Both compact and thin-walled beams are examined to highlight the importance of each term of the three considered expansions. Various loading conditions, including bending, torsion, torsion-bending, and different beam slenderness ratios, are considered. The selected case studies are drawn from existing literature. The accuracy of the models presented is assessed for both displacements and stress components. The results demonstrate that the choice of the most suitable model closely depends on the specific parameters of the individual problem. That is, each structural problem has its own “best” computational models in terms of accuracy versus degree of freedom.

We present a second-order computational multiscale model for heterogeneous porous media, which allows for the scale bridging of transient fluid-flow processes through porous materials with non-separated scales. The formulation connects a homogenized macroscopic scale described by a local theory of grade two with heterogeneous microstructures described by a local theory of grade one. At the macroscale, this leads to $��{�C�}^{�1�}�$-continuity requirements on the macroscopic solution field; at the microscale, we need to take account of constraints on fluctuation fields that require $��H�\left(�\text{div}�\right)�\cap �H�\left(�\text{grad}�\right)�$-conformity of microscopic solutions. Both these challenges are addressed through the development of mixed Hu–Washizu formulations that result in a variationally consistent homogenization framework with minimization structure across scales. We validate the second-order multiscale model by means of fully resolved, direct numerical simulations and provide comparisons with results of first-order FE² simulations. By considering linear Darcy and nonlinear Darcy–Forchheimer flow through two- and three-dimensional porous microstructures, we provide further insights into the framework and associated length-scale effects.

This paper proposes a novel adaptive topology optimization framework that integrates the Virtual Element Method (VEM) with the Material-Field Series Expansion (MFSE). Within the VEM-MFSE framework, we propose a material gradient-driven adaptive strategy, in which elements are refined in regions with higher material density gradients as the MFSE sharpening parameter $��\beta �$ increases. The proposed topology optimization method fully leverages the VEM's advantages of naturally handling arbitrary polygonal elements with hanging nodes. Additionally, MFSE decouples the design variables from the mesh resolution, leading to notable improvements in both the efficiency and convergence of the adaptive optimization. Numerical examples, including the MBB beam benchmark, demonstrate the accuracy and efficiency compared with the Solid Isotropic Material with Penalty (SIMP) method. In addition, an energy-based homogenization scheme is employed for the topology optimization of periodic metamaterial unit cells, where the analytical sensitivity of the effective stiffness with respect to the MFSE coefficient is derived. The results confirm that the adaptive VEM-MFSE method achieves accurate, high-resolution solutions on coarse meshes. For certain problems, the adaptive strategy significantly enhances the convergence behavior of the VEM-MFSE algorithm. Some relevant code is available at https://www.vemhub.com/code.