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This paper presents a stability-preserving model reduction approach for a vibro-acoustic finite element model including poroelastic materials. Most of the research on these systems in the past was conducted in the frequency domain and there were less focus on the stability properties. However, with the increasing of interest in time-domain auralization and virtual sensing, stability-preserving model order reduction is becoming essential for efficient time-domain simulations. The original finite element models for such systems are already well established but not extended to the time domain because of its high computational demand. Therefore, this paper proposes a method to generate stable reduced-order models. The solid displacement–total displacement ( u s − u t ) formulation of the Biot's model is used to describe the poroelastic media. This formulation leads to a set of positive and symmetric system matrices with passive transfer functions. Applying the Kalman–Yakubovich–Popov lemma and the Cholesky/LDL decomposition, this formulation is modified to satisfy the previously proven stability-preserving conditions under one-sided model order reduction. Furthermore, the coupling conditions between the poroelastic media, air, and structure are also investigated. It is finally shown that the stability-preserving property is kept for the coupled models with appropriate variables. The proposed method is verified by several numerical simulations. • A time-domain formulation of Biot's equations is derived. • The time-domain Biot's model is stability-preserving under one-sided model order reduction methods. • The coupling models between the proposed time-domain Biot's model and vibro-acoustic systems are still stability-preserving. • The reduced order model enables efficient time-domain simulations of the vibro-acoustic system with the Biot's model. [ABSTRACT FROM AUTHOR]

In this paper we develop a partition-of-unity construction of the stabilization function required in Nitsche's method, which can be seen as a generalization of the element-wise construction that is widely used in finite element methods. This allows for the use of Nitsche's method within the Partition of Unity Method with a stabilization function that is not simply a constant over the whole boundary. In addition to that, we introduce a patch-aggregation approach designed to avoid arbitrarily large values of the stabilization function and the associated ill-conditioned systems and deteriorated convergence rates. We present numerical results to validate the proposed methods, covering Dirichlet boundary conditions, interface constraints and higher-order problems. These results clearly show that our approach leads to optimal convergence rates. [ABSTRACT FROM AUTHOR]

In this paper, a new framework for large strain of electro-active viscoelastic polymeric beams is developed. The kinematical quantities of beam are derived, and then the constitutive equations of electromechanical beam are developed. To expand the formulation to viscoelastic regime, a generalization of quasi-linear viscoelasticity theory for electo-mechanical deformation is developed and called electro-mechanical quasi-linear viscoelastic (EM-QLV) theory. To deal with highly nonlinear formulation, a finite element formulation in Total Lagrangian framework is derived. Finally, to examine the applicability of the proposed model, several examples are presented and compared with the results in the literature. [ABSTRACT FROM AUTHOR]

This paper solves the discretised multiphase flow equations using tools and methods from machine-learning libraries. The idea comes from the observation that convolutional layers can be used to express a discretisation as a neural network whose weights are determined by the numerical method, rather than by training, and hence, we refer to this approach as Neural Networks for PDEs (NN4PDEs). To solve the discretised multiphase flow equations, a multigrid solver is implemented through a convolutional neural network with a U-Net architecture. Immiscible two-phase flow is modelled by the 3D incompressible Navier–Stokes equations with surface tension and advection of a volume fraction field, which describes the interface between the fluids. A new compressive algebraic volume-of-fluids method is introduced, based on a residual formulation using Petrov–Galerkin for accuracy and designed with NN4PDEs in mind. High-order finite-element based schemes are chosen to model a collapsing water column and a rising bubble. Results compare well with experimental data and other numerical results from the literature, demonstrating that, for the first time, finite element discretisations of multiphase flows can be solved using an approach based on (untrained) convolutional neural networks. A benefit of expressing numerical discretisations as neural networks is that the code can run, without modification, on CPUs, GPUs or the latest accelerators designed especially to run AI codes. • The use of machine learning libraries to solve discretised systems of PDEs exactly. • The first application of this idea to multiphase flows with interface-capturing methods. • The code can run, with no modification, on CPUs and GPUs. • It also runs, with no modification, on the latest energy-efficient AI processors. • A new compressive advection scheme sharpens the interface between the fluids. [ABSTRACT FROM AUTHOR]

This paper presents an efficient solution for designing the topology of structures that can withstand dynamic loads. The method, called image/stereolithography (STL)-based adaptive scaled boundary finite element (SBFE), is a novel approach to topology optimization (TO) that is particularly effective for designing two/three-dimensional structures. The SBFE-based TO algorithm adopts a bi-evolutionary structural optimization (BESO) approach, which utilizes image/STL-based procedures to automatically adapt meshes and minimize mechanical compliance under varying volume fractions in the time domain. A key advantage of this approach is that it uses images of TO responses, rather than specific error functions, to guide the automatic adaptive mesh schemes. The method also incorporates a quadtree/octree mesh refinement to address the problem of hanging nodes, ensuring fast mesh convergence during the optimization process. Furthermore, a high-order implicit time integration technique provides a close approximation of equivalent static loads, allowing for the accurate mapping of displacement fields under actual dynamic responses at each time step. The effectiveness of this method is demonstrated through a variety of numerical examples and benchmarks. • SBFE-based TO algorithm adopts image/STL procedures for automated mesh adaptation. • Multiple dynamic loadings applied in optimization result in various optimal layouts. • Implicit time integration based on Padé expansions is employed in elastodynamic analysis. • Objective function, weighted summation of compliances near critical peak points, reduces CPU time. • Optimal results undergo further smoothing using Fusion 360, a commercial software. [ABSTRACT FROM AUTHOR]

Accurate material description is crucial to achieve high-quality results in computational analysis software. Phenomenological constitutive laws generalize the material behaviour observed in simple mechanical tests. The resulting empirical expressions contain parameters that need to be calibrated through an inverse optimization process. Advancements in Digital Image Correlation (DIC) techniques have enabled the extraction of non-uniform multi-axial displacement fields, facilitating the development of heterogeneous mechanical specimens and unlocking access to richer material behaviour data. Despite these advancements, constitutive models are still limited by mathematical formulations and biases due to simplifying assumptions. Artificial Neural Networks (ANNs), as universal function approximators, could potentially drive a paradigm shift in the field. ANNs do not require explicit pre-formulations, hence avoiding the need for identifying unknown parameters. Moreover, ANNs are able to implicitly learn patterns purely from data, allowing to circumvent the biases induced by the simplifying assumptions of first-order principles. However, the training of ANNs for implicit constitutive modelling is not straightforward, given the impossibility of obtaining direct measurements of stress. This paper explores the integration of Recurrent Neural Networks (RNNs) with the Virtual Fields Method (VFM) for material modelling. This approach uses global force and displacement data to indirectly train the neural network, with the equilibrium being evaluated globally through the VFM loss function. The proposed method is (i) computationally efficient by not requiring Finite Element Analysis (FEA), (ii) compatible with both full-field measurements and numerically generated data, and (iii) able to handle experimental boundary conditions. • A novel method integrates RNNs with the VFM for implicit material modelling. • The RNN is trained without labels, using only displacement/strain and global force. • The approach is real-test oriented, compatible with both DIC and numerical data. • The methodology is independent of the boundary conditions. • The method is computationally more efficient than FEA indirect training. [ABSTRACT FROM AUTHOR]

Despite substantial advancements in optimizing structural designs for stiffness, the field of design against fracture is still in its early stages. This paper introduces a fundamental and standardized paradigm for structural design against brittle fracture, achieved through the minimization of the energy release rate G within the framework of linear elastic fracture mechanics (LEFM). By leveraging the Griffith criterion expression and the total potential energy in LEFM, we propose an efficient approach for structural optimization that enhances the toughness of architected materials. The energy release rate G is evaluated a synergistic application of finite element analysis and finite difference methods, yielding a precision of 0.6% compared to benchmarks in published handbooks. We demonstrate significantly reduced G values compared to standard stiffness and stress-oriented topology optimization, which directly contributes to increased toughness. To comprehensively analyze the fracture process encompassing damage, crack initiation, propagation, and failure, we employ post-processing phase field fracture modeling on the optimally designed architected materials. By performing multi-objective optimization of the stiffness and energy release rate G , the toughness and extreme load-bearing capacity can be simultaneously improved. The resulting perforated architecture effectively outperforms homogeneous structures in both toughness and peak load. Additionally, we extend our analysis to various configurations, such as non-center cracks, pure shear loading, and adherence to the standard 3D compact tension ASTM E-399-72. The designed structures are additively manufactured and experimentally validated, demonstrating a toughness increase of over 10 times compared to stress-based design. These contributions offer promising prospects for advancing the design of future engineering structures and materials with a favorable balance of critical fracture resistance characteristics. [ABSTRACT FROM AUTHOR]

The a posteriori error analysis of the classical Argyris finite element methods dates back to 1996, while the optimal convergence rates of associated adaptive finite element schemes are established only very recently in 2021. It took a long time to realize the necessity of an extension of the classical finite element spaces to make them hierarchical. This paper establishes the novel adaptive schemes for the biharmonic eigenvalue problems and provides a mathematical proof of optimal convergence rates towards a simple eigenvalue and numerical evidence thereof. This makes the suggested algorithm highly competitive and clearly justifies the higher computational and implementational costs compared to low-order nonconforming schemes. The numerical experiments provide overwhelming evidence that higher polynomial degrees pay off with higher convergence rates and underline that adaptive mesh-refining is mandatory. Five computational benchmarks display accurate reference eigenvalues. [ABSTRACT FROM AUTHOR]

This paper presents an extension of the discrete element method using a phase-field formulation to incorporate grain shape and its evolution. The introduction of a phase variable enables an effective representation of grain geometry and facilitates the application of physical laws, such as chemo-mechanical couplings, for modeling shape changes. These physical laws are solved numerically using the finite element method coupled in a staggered scheme to the discrete element model. The efficacy of the proposed Phase-Field Discrete Element Model (PFDEM) is demonstrated through its ability to accurately capture the real grain shape in a material subjected to dissolution only and compute the stress evolution. It is then applied to model the phenomenon of pressure solution involving dissolution and precipitation in granular materials at the microscale and enables to reproduce the creep response observed experimentally. This framework contributes to the enhanced understanding and simulation of complex behaviors in granular materials and sedimentary rocks for many geological processes like diagenesis or earthquake nucleation. • Coupling between PF and DEM to incorporate grain shape and its evolution. • The grain shape influes the behavior to an oedometer test under acid injection. • Model the phenomenon of pressure solution at the microstructural level. [ABSTRACT FROM AUTHOR]

The paper deals with the rebound of an elastic solid off a rigid wall of a container filled with an incompressible Newtonian fluid. Our study focuses on a collision-free bounce, meaning a rebound without topological contact between the elastic solid and the wall. This has the advantage of omitting any artificial bouncing law. In order to capture the contact-free rebound for very small viscosities an adaptive numerical scheme is introduced. The here-introduced scheme is based on a Glowinski time scheme and a localized arbitrary Lagrangian–Eulerian map on finite elements in space. The absence of topological contact requires that very thin liquid channels are solved with sufficient accuracy. It is achieved via newly developed geometrically driven adaptive strategies. Using the numerical scheme, we present here a collection of numerical experiments. A rebound is simulated in the absence of topological contacts. Its physical relevance is demonstrated as, with decreasing viscosities, a free rebound in a vacuum is approached. Further, we compare the dynamics with a second numerical scheme; a here-introduced adaptive purely Eulerian level-set method. The scheme produced the same dynamics for large viscosities. However, as it requires a much higher computational cost, small viscosities cannot be reached by this method. The experiments allow for a better understanding of the effect of fluids on the dynamics of elastic objects. Several observations are discussed, such as the amount of elastic and/or kinetic energy loss or the precise connection between the fluid pressure and the rebound of the solid. [Display omitted] • Contactless rebound of elastic solids in fluids. • Viscosity convergence of (contactless) rebound to vacuum case. • Updated arbitrary Lagrangian–Eulerian method with re-meshing. • A new re-meshing strategy based on the solution of the Eikonal equation. [ABSTRACT FROM AUTHOR]

The computational analysis of nanophotonic devices is usually carried out via the standard finite element method (FEM). However, FEM requires meshes that are fitted to the devices' boundaries, so making changes to the geometry (and thus the mesh) results in an inefficient process at best. Such an approach is therefore at odds when conducting design, which requires the analysis of multiple device geometries until reaching a satisfactory solution. Computational design tools such as topology optimization are often used, but the use of density-based representations of geometry inevitably leads to other issues—e.g., pixelized fuzzy boundaries with "gray material" (that does not correspond to dielectric nor vacuum) have an adverse effect on the devices' interaction with electromagnetic waves. In this paper we propose an interface-enriched generalized finite element method (IGFEM) for the analysis of two-dimensional electromagnetic scattering and eigenvalue problems. IGFEM enables the use of finite element meshes that are completely decoupled from the problem's geometry. The analysis procedure is further coupled to a level set description of topology, resulting in a versatile enriched approach to topology optimization; this level set-based interface-enriched topology optimization procedure is devoid of the issues mentioned above regarding density-based methods, and yields crisp "black-and-white" designs that are devoid of jagged fuzzy edges. We first demonstrate that the analysis procedure achieves the same convergence rate as that of standard FEM using geometry-fitted meshes. We then compare the convergence properties of IGFEM with Nitsche's method on a problem containing an embedded straight interface. Finally, we conduct topology optimization for designing both a 2-D metalens and a 2-D reflector, maximizing their ability to focus light onto a target point. [Display omitted] • Interface-enriched GFEM with nodal elements allows to decouple mesh from geometry. • Nodal enrichments allow for straightforward implementation in displacement-based FEM codes. • IGFEM-based analysis has optimal error convergence in L 2 norm for smooth problems. • Photonic crystal band structures can be computed accurately with IGFEM-based analysis. • Enriched level set-based topology optimization yields black-and-white smooth designs. [ABSTRACT FROM AUTHOR]

• The added degree of freedom method (ADM) is proposed to solve the absorbing boundary problem. • The principle of a mass damper is employed in the ADM to achieve the attenuation of stress waves. • The ADM can absorb one-dimensional and two-dimensional elastic waves across a broad range of frequencies. Absorbing boundaries are essential in engineering simulations, especially for elastodynamic problems, to ensure accuracy, reliability and numerical stability. In this paper, the added degree of freedom method (ADM) is developed as a novel approach to address absorbing boundary challenges in finite element analysis. In ADM, additional degrees of freedom (DOFs) are introduced within the absorbing domain to attenuate outgoing elastic waves. For the proposed ADM to implement the absorbing boundary, the stiffness and mass properties of both the added DOFs and conventional DOFs within the absorbing domain are adjusted. This adjustment aims to reduce the propagation speed of elastic waves within the medium, and to prolong the duration of interaction between elastic waves and the surrounding medium. Consequently, the vibrations of nodes can be effectively attenuated by applying damping forces to the added DOFs. The numerical results show that the ADM has the ability to absorb one-dimensional and two-dimensional elastic waves across a broad range of frequencies. Therefore, the proposed ADM offers an innovative solution for modeling absorbing boundaries in various scientific and engineering applications, addressing the challenges of simulating wave propagation within finite computational domains. [ABSTRACT FROM AUTHOR]

This paper investigates the numerical approximation of a two-phase Hele-Shaw fluid flow system, which consists of a phase-field Cahn–Hilliard model coupled with the Darcy flow, confined on a complex curved surface. To avoid geometric errors caused by the surface approximation and additional approximation errors introduced by numerical methods, for spatial discretization, we consider the recently developed strategy of subdivision-based isogeometric analysis (IGA), where we can accurately represent complex surfaces of arbitrary topology due to its high smoothness and hierarchical refinement properties. By incorporating multiple methods of temporal discretizations, including the Explicit-Invariant-Energy-Quadratization (EIEQ) approach for linearizing nonlinear potentials, the Zero-Energy-Contribution (ZEC) method for decoupling of nonlinear couplings, and the projection method for decoupling of linear couplings between the fluid velocity and pressure, and the finite-element method based on IGA for spatial discretizations, we arrive at a fully discrete scheme that possesses desirable properties such as decoupling, linearity, second-order accuracy in time, and unconditional energy stability. At each time step, only several elliptic equations with constant coefficients are needed to be solved. We also present the rigorous proof of unconditional energy stability, accompanied by numerous numerical examples to verify the stability and accuracy of the numerical scheme, such as the Saffman–Taylor fingering instability occurring on various complex curved surfaces induced by rotational forces. [ABSTRACT FROM AUTHOR]

The Scaled Boundary Finite Element Method (SBFEM) can be viewed as a semi-analytical approach, which has been successfully applied to singular mechanical problems with vanishing force terms. Interpreted in the context of composite Duffy approximations based on star-shape polytopes (S -elements), and using an operator-adapted principle, the shape functions are determined by imposing an energy-orthogonality constraint. In the presence of non-vanishing forcing terms, the SBFEM approximation converges at sub-optimal rates. In the current paper, a strategy is proposed to overcome this drawback by the enrichment of the classic SBFEM spaces with energy-orthogonal bubble Duffy functions. The orthogonality property leads to a decoupling of the approximate solutions into components of boundary trace and interior (bubble) functions. The bubble functions can be solved separately, thus reducing the computational cost. A priori energy-error estimates are provided demonstrating optimal rates of convergence. The convergence rates are confirmed by two and three-dimensional numerical verification tests. • Operator-adapted SBFEM approximations in the presence of body force are proposed. • A priori error estimates for SBFEM with bubble enrichment are provided. • Optimal convergence rates for Poisson and elasticity problems are numerically verified. • The SBFEM bubble-enriched approximations hold energy-orthogonal properties. • The orthogonality property leads to a reduction in the computational cost. [ABSTRACT FROM AUTHOR]

Numerical prediction of melt pool morphology and temperature distribution of thermomechanical processes (welding, additive manufacturing) plays an important role in understanding the relationships between process parameters and the quality of manufactured parts. The heat conduction models are limited in their predictability because the transport phenomena relevant to the melt pool dynamic are ignored. In contrast, the multiphysics model allows consideration of the heat transfer, free surface evolution, and transport phenomena in the melt pool, leading to more accurate predictions. However, the computational cost of multiphysics models makes them impractical for part-scale simulations. In this paper, a local moving thermal-fluid framework based on the finite element method is presented, which allows solving the thermal-fluid problem only in the small moving zone containing the melt pool and the heat transfer problem in the rest part. Therefore, this new proposed moving thermal-fluid (MTF) framework is predictive as well as the full thermal-fluid model, while it is much more efficient than a full thermal-fluid model since there are much fewer degrees of freedom (DOF) to solve. Finally, numerical validation tests in welding and additive manufacturing simulations highlight its computational efficiency and high-fidelity, and a relative error of less than 2.6% was observed in predicting melting pool dimensions compared to full thermal-fluid model. In the part-scale simulation of laser welding benchmark, the MTF model takes only 62 h in place of about three weeks required by a full thermal-fluid model. • A computational efficient local moving thermal-fluid framework is proposed. • The thermal-fluid problem is solved only in the small zone containing the melt pool. • The new model is applied for modeling welding and additive manufacturing processes. • The accuracy and efficiency of new approach is verified against numerical references. [ABSTRACT FROM AUTHOR]

This paper presents an approach to the numerical estimation of effective properties of highly porous materials based on the scaled boundary finite element method (SBFEM). The latter can be formulated on quadtree meshes with hanging nodes and thus facilitates the efficient mesh generation and analysis of a large number of randomly created samples. To generate the corresponding Representative Volume Elements (RVEs), an improved Random Sequential Addition (RSA) method with overlap is used. Moreover, an alternative image-based digital matrix method for generating random assemblies of circular particles of the same size is proposed. In addition, a modified smoothing algorithm to treat the staircase boundaries present in a quadtree mesh is developed in order to accurately and efficiently capture the sinter-neck features of overlapping circular particles. The proposed image-based mesh generation and analysis procedure is used to evaluate the relationship between microstructural parameters and effective Young's modulus considering a large number of samples. In this context, a new dimensionless morphological parameter N f c is proposed. The latter represents the morphology and connectivity of voids and is shown to be a useful indicator with respect to the effective elastic properties of highly porous ceramic matrix material. • Scaled boundary finite element method for homogenization analysis of highly porous materials. • Model generation method for large number of Representative Volume Elements (RVE) of porous ceramic matrix material. • New automatic quadtree meshing and smoothing algorithm with high accuracy and efficiency for complex structures. • New dimensionless parameter that exhibits a stronger correlation with the material's geometric properties and effective elastic properties than porosity. [ABSTRACT FROM AUTHOR]

This paper introduces an extended tensor decomposition (XTD) method for model reduction. The proposed method is based on a sparse non-separated enrichment to the conventional tensor decomposition, which is expected to improve the approximation accuracy and the reducibility (compressibility) in highly nonlinear and singular cases. The proposed XTD method can be a powerful tool for solving nonlinear space–time parametric problems. The method has been successfully applied to parametric elastic–plastic problems and real time additive manufacturing residual stress predictions with uncertainty quantification. Furthermore, a combined XTD-SCA (self-consistent clustering analysis) strategy is presented for multi-scale material modeling, which enables real time multi-scale multi-parametric simulations. The efficiency of the method is demonstrated with comparison to finite element analysis. The proposed method enables a novel framework for fast manufacturing and material design with uncertainties. • Novel eXtended Tensor Decomposition (XTD) based nonlinear model reduction method. • Sparse non-separated enrichment can improve the accuracy, convergence, and robustness of model reduction methods. • XTD has been applied to nonlinear elastic–plastic problems for stress predictions in additive manufacturing. • The combined XTD-SCA (self-consistent clustering analysis) can enable fast multiscale material modeling and design under uncertainty. [ABSTRACT FROM AUTHOR]

This paper develops an adaptive algorithm for the Generalized Finite Element Method with global–local enrichment (GFEM gl) for transient multiscale PDEs. The adaptive algorithm detects a subset of global nodes with trivial enrichments, which are exactly or close to linearly dependent from the underlying coarse FEM basis, at each time step, and then removes them from the global system. It is based on the calculation of the ratio between the largest and smallest singular values of small sub-matrices extracted from the global system of equations which introduces little overhead over the non-adaptive GFEM gl for transient PDEs. Compared to existing adaptive multiscale approaches, where either an a-posterior error estimate, a change in physical quantities, or a local problem residual is calculated, the proposed approach provides an innovative framework based on singular values. The proposed approach is shown to be robust for solving advection–diffusion problems that require detecting initial conditions with spikes and capturing moving/morphing/merging mass plumes in heterogeneous media and non-uniform flow with sharp fronts. Specific examples are provided for applications in groundwater contaminant and heat dissipation. The accuracy of the proposed adaptive GFEM gl closely matches reference fine-scale FEM solutions in the L ∞ norm. [ABSTRACT FROM AUTHOR]

An efficient concurrent topology optimization method based on the Extended Multiscale Finite Element Method (EMsFEM) is proposed to design the hierarchical structure with multiple substructures in this paper. Firstly, the mathematical description model for the topology of hierarchical structures and the interpolation model for material are established at both the macro and substructural levels. Then the concurrent topology optimization formulation for hierarchical structures is proposed based on EMsFEM. After that, the sensitivity analysis scheme of the objective function and constraint functions is given, based on which the Method of Moving Asymptotes (MMA) is employed to update design variables. The assumption of separation of length scales is not required in the proposed method, which avoids the disconnection between adjacent substructures. Besides, without limiting the topology and spatial distribution of substructures in advance, the concurrent topology optimization for hierarchical structures with multiple substructures can be realized using this method. The provided numerical examples verify the correctness and effectiveness of the proposed sensitivity analysis method as well as the feasibility, effectiveness and scalability of the proposed concurrent topology optimization method for hierarchical structures. Finally, the influence of the initial substructure configurations and the number of substructure types on the final configuration and mechanical performance of hierarchical structures are explored with a series of numerical examples. • An efficient concurrent topology optimization method for designing hierarchical structures with multiple substructures is proposed based on the Extended Multiscale Finite Element Method (EMsFEM). • Without aforehand limitation on substructures' topologies and their spatial distribution, the proposed method can simultaneously optimize the topologies and spatial distribution of substructures in hierarchical structures. • The proposed method is free of the assumption of separation of length scales, which not only avoids the connection problem between adjacent substructures but also makes the manufacturing of optimized hierarchical structures of practical significance. [ABSTRACT FROM AUTHOR]

In this paper, an efficient formulation for the topology optimization of extruded beams is developed considering linearized buckling, displacement, and stress constraints. The proposed method relies on a computationally efficient extended\generalized finite element (XFEM\GFEM) formulation for the forward problem, recently developed for the analysis of beams. In this method, a coarse 3D finite element (FE) mesh is enhanced with global enrichment functions, enabling a substantial reduction in the number of degrees of freedom with comparable accuracy to a traditional 3D FE fine mesh. The material distribution at the cross-section is optimized utilizing the material density approach with the Solid Isotropic Material with Penalization (SIMP) method. Our objective is to minimize the total weight of the beam, subject to performance constraints including buckling, displacement, and stress. The forward problem enables the application of general loading and boundary conditions. Thus, both buckling and lateral–torsional buckling are automatically considered. The optimization problem is solved using a gradient-based optimization approach, with analytical gradients derived by using the adjoint variable method. Several numerical examples are presented to investigate the accuracy and efficiency of the proposed formulation. A parametric study showed that the optimized cross-section is strongly dependent on the length and permissible buckling load. In addition, it is shown that a simultaneous formulation leads to significantly different designs, with different structural behavior, compared to results obtained by considering only a subset of the performance constraints. Finally, we verify the model on a beam warping problem under pure torsion, in which the cross-section optimization converges to a known analytical solution. Compared with traditional FEM, two orders of magnitude in terms of execution time are reported in favor of the proposed XFEM\GFEM approach. • A new approach for the topology optimization of extruded beams is proposed. • The forward problem is solved using an efficient numerical scheme relying on the extended finite element method. • The approach simultaneously considers buckling, stress, and displacement constraints. • We show that the beam's length significantly affects the optimal shape. • The proposed XFEM\GFEM approach shows significant time savings compared with 3D FE. [ABSTRACT FROM AUTHOR]

This paper proposes an invariant-domain preserving approximation technique for nonlinear conservation systems that is high-order accurate in space and time. The algorithm mixes a high-order finite element method with an invariant-domain preserving low-order method that uses the closest neighbor stencil. The construction of the flux of the low-order method is based on an idea from Abgrall et al. (2017). The mass flux of the low-order and the high-order methods are identical on each finite element cell. This allows for mass preserving and invariant-domain preserving limiting. [ABSTRACT FROM AUTHOR]