L-shaped Domain Research Articles

Finite element simulations to explore thermofluidic transport mechanism in permeable medium by using Darcy Brinkman-Forchheimer model

Diversified utilizations of permeable domains are found in cooling systems, gas recovery, and insulation engineering. Therefore, this research focused on investigating the heat transport aspects imparted by the buoyantly driven flow of a viscous fluid saturated in a permeable L-shaped enclosure. Darcy Brinkman-Forchheimer model is considered in the present pagination to envision permeability aspects. In addition, entropy generation is considered to encapsulate the disordered form of the system. In this study, an l-shaped domain with heated steps and insulated horizontal boundaries was entertained, while the vertical walls are kept cold. Non-persistent buoyancy forces are employed to elucidate the impact of the gravitational forces. The mathematical formulation of the problem is examined in the form of a dimensionless coupled partial differential system after normalizing the variables. Subsequently, the Galerkin finite element method was implemented using a commercial multigrid software (COMSOL) to find a solution to the constructed problem. The influence of the included parameters on the related distributions and quantities of interest in a comparative sense is evaluated using sketches and tables. A comparison of the results along with a grid-independent test is also performed to ensure the reliability of the computational procedure and computed outcomes. It is inferred that the total entropy and averaged Nusselt number increase as the Darcy number increases by 21.39% and 10.76%, respectively, whereas the Bejan number exhibits the opposite trend. Raising the aperture ratio of the enclosure results in a reduction of the heat flux coefficient, and the ECOP also indicates a reduction in entropy.

Highly efficient NURBS-based isogeometric analysis for coupled nonlinear diffusion–reaction equations with and without advection

Nonlinear diffusion–reaction systems model a multitude of physical phenomena. A common situation is biological development modeling where such systems have been widely used to study spatiotemporal phenomena in cell biology. Systems of coupled diffusion–reaction equations are usually subject to some complicated features directly related to their multiphysics nature. Moreover, the presence of advection is source of numerical instabilities, in general, and adds another challenge to these systems. In this study, we propose a NURBS-based isogeometric analysis (IgA) combined with a second-order Strang operator splitting to deal with the multiphysics nature of the problem. The advection part is treated in a semi-Lagrangian framework and the resulting diffusion–reaction equations are then solved using an efficient time-stepping algorithm based on operator splitting. The accuracy of the method is studied by means of a advection–diffusion–reaction system with analytical solution. To further examine the performance of the new method on geometries more general than rectangles (e.g., L-shaped domains and parts of annuli), the well-known Schnakenberg–Turing problem is considered with and without advection. Finally, a Gray–Scott system on a circular domain is also presented. The results obtained demonstrate the efficiency of our new algorithm to accurately reproduce the solution in the presence of complex patterns on more complicated geometries. Moreover, the new method clarifies the effect of geometry on Turing patterns.

Extraction of stress intensity factors of biharmonic equations with free boundary conditions on crack faces (III): SIF of a thin plate subjected to the bending moment

The plate bending problems in the Kirchhoff–Love model can be reduced to biharmonic equations. In the previous papers (Kim et al., 2020; Kim et al., 2022), we derived stress intensity factors (SIF) extraction formulas of a biharmonic equation Δ2u=f in a non-convex (cracked, L-shaped) domain with various boundary conditions such as clamped, simply connected, free, and mixed. In previous papers, we could not show a general SIF formula related to Poisson’s ratio ν contained in the free boundary condition. Moreover, the previous SIF extraction formula obtained for a given Poisson’s ratio requires cumbersome line integrals. In this paper, removing the line integrals in the extraction formulas, we show improved SIF extraction formulas applicable to biharmonic equations with free boundary conditions. Thus, the new, improved SIF extraction formulas for free boundary conditions become more straightforward and can be applied to plate bending problems.

The eXtended virtual element method for elliptic problems with weakly singular solutions

This paper introduces a novel eXtended virtual element method, an extension of the conforming virtual element method. The X-VEM is formulated by incorporating appropriate enrichment functions in the local spaces. The method is designed to handle highly generic enrichment functions, including singularities arising from fractured domains. By achieving consistency on the enrichment space, the method is proven to achieve arbitrary approximation orders even in the presence of singular solutions. The paper includes a complete convergence analysis under general assumptions on mesh regularity, and numerical experiments validating the method’s accuracy on various mesh families, demonstrating optimal convergence rates in the L2- and H1-norms on fractured or L-shaped domains.

Data-based adaptive refinement of finite element thin plate spline

The thin plate spline, as introduced by Duchon, interpolates a smooth surface through scattered data. It is computationally expensive when there are many data points. The finite element thin plate spline (TPSFEM) possesses similar smoothing properties and is efficient for large data sets. Its efficiency is further improved by adaptive refinement that adapts the precision of the finite element grid. Adaptive refinement processes and error indicators developed for partial differential equations may not apply to the TPSFEM as it incorporates information about the scattered data. This additional information results in features not evident in partial differential equations. An iterative adaptive refinement process and five error indicators were adapted for the TPSFEM. We give comprehensive depictions of the process in this article and evaluate the error indicators through a numerical experiment with a model problem and two bathymetric surveys in square and L-shaped domains.

Adaptive Deep Fourier Residual method via overlapping domain decomposition

The Deep Fourier Residual (DFR) method is a specific type of variational physics-informed neural network (VPINN). It provides a robust neural network-based solution to partial differential equations (PDEs). The DFR strategy is based on minimizing the dual norm of the weak residual of a PDE, which is equivalent to minimizing the energy norm of the error. To compute the dual norm of the weak residual, the DFR method employs an orthonormal spectral basis of the test space, known for rectangles or cuboids for multiple function spaces.In this work, we extend the DFR method with ideas of traditional domain decomposition (DD). This enables two improvements: (a) to solve problems in more general polygonal domains and (b) to develop an adaptive refinement technique in the test space using a Döfler marking algorithm. In the former case, we retain the desirable equivalence between the employed loss function and the H1 error under non-restrictive assumptions, numerically demonstrating adherence to explicit bounds in the case of the L-shaped domain problem. In the latter, we show how refinement strategies lead to potentially significant improvements against a reference, classical DFR implementation with a test function space of significantly lower dimensionality, allowing us to better approximate singular solutions at a more reasonable computational cost.

Wang tiles enable combinatorial design and robot-assisted manufacturing of modular mechanical metamaterials

In this paper, we introduce a novel design paradigm for modular architectured materials that allows for spatially nonuniform designs from a handful of building blocks, which can be robotically assembled for efficient and scalable production. The traditional, design-limiting periodicity in material design is overcome by utilizing Wang tiles to achieve compatibility among building blocks. We illustrate our approach with the design and manufacturing of an L-shaped domain inspired by a scissor-like soft gripper, whose internal module distribution was optimized to achieve an extreme tilt of a tip of the gripper’s jaw when the handle part was uniformly compressed. The geometry of individual modules was built on a 3 × 3 grid of elliptical holes with varying semi-axes ratios and alternating orientations. We optimized the distribution of the modules within the L-shaped domain using an enumeration approach combined with a factorial search strategy. To address the challenge of seamless interface connections in modular manufacturing, we produced the final designs by casting silicone rubber into modular molds automatically assembled by a robotic arm. The predicted performance was validated experimentally using a custom-built, open-hardware test rig, Thymos, supplemented with digital image correlation measurements. Our study demonstrates the potential for enhancing the mechanical performance of architectured materials by incorporating nonuniform modular designs and efficient robot-assisted manufacturing.

A Reissner–Mindlin plate formulation using symmetric Hu-Zhang elements via polytopal transformations

In this work we develop new finite element discretisations of the shear-deformable Reissner–Mindlin plate problem based on the Hellinger–Reissner principle of symmetric stresses. Specifically, we use conforming Hu-Zhang elements to discretise the bending moments in the space of symmetric square integrable fields with a square integrable divergence M∈HZ⊂Hsym(Div). The latter results in highly accurate approximations of the bending moments M and in the rotation field being in the Lebesgue space ϕ∈[L2]2, such that the Kirchhoff–Love constraint can be satisfied for t→0. In order to preserve optimal convergence rates across all variables for the case t→0, we present an extension of the formulation using Raviart–Thomas elements for the shear stress q∈RT⊂H(div).We prove existence and uniqueness in the continuous setting and rely on exact complexes for inheritance of well-posedness in the discrete setting.This work introduces an efficient construction of the Hu-Zhang base functions on the reference element via the polytopal template methodology and Legendre polynomials, making it applicable to hp-FEM. The base functions on the reference element are then mapped to the physical element using novel polytopal transformations, which are suitable also for curved geometries.The robustness of the formulations and the construction of the Hu-Zhang element are tested for shear-locking, curved geometries and an L-shaped domain with a singularity in the bending moments M. Further, we compare the performance of the novel formulations with the primal-, MITC- and recently introduced TDNNS methods.

Physics-informed neural networks with residual/gradient-based adaptive sampling methods for solving partial differential equations with sharp solutions

We consider solving the forward and inverse partial differential equations (PDEs) which have sharp solutions with physics-informed neural networks (PINNs) in this work. In particular, to better capture the sharpness of the solution, we propose the adaptive sampling methods (ASMs) based on the residual and the gradient of the solution. We first present a residual only-based ASM denoted by ASM I. In this approach, we first train the neural network using a small number of residual points and divide the computational domain into a certain number of sub-domains, then we add new residual points in the sub-domain which has the largest mean absolute value of the residual, and those points which have the largest absolute values of the residual in this sub-domain as new residual points. We further develop a second type of ASM (denoted by ASM II) based on both the residual and the gradient of the solution due to the fact that only the residual may not be able to efficiently capture the sharpness of the solution. The procedure of ASM II is almost the same as that of ASM I, and we add new residual points which have not only large residuals but also large gradients. To demonstrate the effectiveness of the present methods, we use both ASM I and ASM II to solve a number of PDEs, including the Burger equation, the compressible Euler equation, the Poisson equation over an L-shape domain as well as the high-dimensional Poisson equation. It has been shown from the numerical results that the sharp solutions can be well approximated by using either ASM I or ASM II, and both methods deliver much more accurate solutions than the original PINNs with the same number of residual points. Moreover, the ASM II algorithm has better performance in terms of accuracy, efficiency, and stability compared with the ASM I algorithm. This means that the gradient of the solution improves the stability and efficiency of the adaptive sampling procedure as well as the accuracy of the solution. Furthermore, we also employ the similar adaptive sampling technique for the data points of boundary conditions (BCs) if the sharpness of the solution is near the boundary. The result of the L-shape Poisson problem indicates that the present method can significantly improve the efficiency, stability, and accuracy.

Conformal Mapping of an L-Shaped Domain in Analytical Form

Conformal Mapping of an L-Shaped Domain in Analytical Form

A combined multiscale finite element method based on the LOD technique for the multiscale elliptic problems with singularities

In this paper, we construct a combined multiscale finite element method (MsFEM) using the Local Orthogonal Decomposition (LOD) technique to solve the multiscale problems which may have singularities in some special portions of the computational domain. For example, in the simulation of steady flow transporting through highly heterogeneous porous media driven by extraction wells, the singularities lie in the near-well regions. The basic idea of the combined method is to utilize the traditional finite element method (FEM) directly on a fine mesh of the problematic part of the domain and using the LOD-based MsFEM on a coarse mesh of the other part. The key point is how to define local correctors for the basis functions of the elements near the coarse and fine mesh interface, which require meticulous treatment. The proposed method takes advantages of the traditional FEM and the LOD-based MsFEM, which uses much less DOFs than the standard FEM and may be more accurate than the LOD-based MsFEM for problems with singularities. The error analysis is carried out for highly varying coefficients, without any assumptions on scale separation or periodicity. Numerical examples with periodic and random highly oscillating coefficients, as well as the multiscale problems on the L-shaped domain, and multiscale problems with high-contrast channels or well-singularities are presented to demonstrate the efficiency and accuracy of the proposed method.

New Numerical Approach for the Steady-State Navier–Stokes Equations with Corner Singularity

The steady-state nonlinear problem governing the Newtonian flow of an incompressible viscous fluid in convection form in L-shaped domain is considered. The solution as [Formula: see text]-generalized one in weighted sets is determined. Unlike the classical definition of the generalized solution in a weak formulation, which has a symmetric structure, our setting has an asymmetric one so that a special inf-sup condition in weighted sets is proved. A new weighted finite element scheme for an approximate [Formula: see text]-generalized solution is constructed. A converging iterative method to solve the sequence of linear problems with block preconditioning of their matrices is introduced. The results of numerical simulations of several tests have shown an advantage over the standard methods so that an approximate [Formula: see text]-generalized solution tends to the exact one with a rate almost twice in order exceed, relative to the grid step, than classical approaches. The result is achieved without using mesh refined in the neighborhood of a reentrant corner.

POLYLLA: polygonal meshing algorithm based on terminal-edge regions

This paper presents an algorithm to generate a new kind of polygonal mesh obtained from triangulations. Each polygon is built from a terminal-edge region surrounded by edges that are not the longest-edge of any of the two triangles that share them. The algorithm is termed Polylla and is divided into three phases. The first phase consists of labeling each edge of the input triangulation according to its size; the second phase builds polygons (simple or not) from terminal-edges regions using the label system; and the third phase transforms each non simple polygon into simple ones. The final mesh contains polygons with convex and non convex shape. Since Voronoi-based meshes are currently the most used polygonal meshes, we compare some geometric properties of our meshes against constrained Voronoi meshes. Several experiments were run to compare the shape and size of polygons, the number of final mesh points and polygons. For the same input, Polylla meshes contain less polygons than Voronoi meshes and the algorithm is simpler and faster than the algorithm to generate constrained Voronoi meshes. Finally, we have validated Polylla meshes by solving the Laplace equation on an L-shaped domain using the virtual element method (VEM). We show that the numerical performance of the VEM using Polylla meshes and Voronoi meshes is similar.

A Computational Strategy for Code- and Mesh-Agnostic Nonlinear Global–Local Analysis

Global, coarse mesh and local, refined mesh modeling strategy is a common solution approach for domains spanning many orders in length scale. In this article, a nonintrusive computational strategy for iterative solution to the nonlinear behavior in the subdomains is proposed. To estimate the residual force on the interface connecting the subdomains with nonmatching discretizations, variational principles with an intermediate framework are proposed to assure satisfaction of virtual work principle in the subdomains and to assure displacement/traction compatibility at the interface. To map nodal force from one subdomain to the other, both global Lagrange multiplier (GLM) and local Lagrange multiplier (LLM) methods are discussed. Quasi-Newton iterative updates are used to correct displacements on the interface connecting the different subdomains until equilibrium is reached. A symmetric rank 1 update as well as the Broyden–Fletcher–Goldfarb–Shanno (BFGS) update are considered for accelerating the solution convergence. The developed method is suitable for the modular construction of submodels with nonmatching discretizations and even allows coupling between subdomains analyzed using different (commercial or custom) codes. The iterative global–local coupling strategy is validated on several examples, including an L-shaped domain with local nonlinearity and a rectangular plate with a propagating crack. Another example illustrating the coupling of domains independently analyzed using commercial finite element codes ANSYS and ABAQUS is next demonstrated. Finally, the method is demonstrated to analyze semiconductor chip assemblies where plastic ratcheting of the interconnect line causes passivation coating fracture.

An adjoint-based super-convergent Galerkin approximation of eigenvalues

We present a new method for computing high-order accurate approximations of eigenvalues defined in terms of Galerkin approximations. We consider the eigenvalue as a non-linear functional of its corresponding eigenfunction and show how to extend the adjoint-based approach proposed in Cockburn and Wang (2017) [14], to compute it. We illustrate the method on a second-order elliptic eigenvalue problem. Our extensive numerical results show that the approximate eigenvalues computed by our method converge with a rate of 4k+2 when tensor-product polynomials of degree k are used for the Galerkin approximations. In contrast, eigenvalues obtained by standard finite element methods such as the mixed method or the discontinuous Galerkin method converge with a rate at most of 2k+2. We present numerical results which show the performance of the method for the classic unit square and L-shaped domains, and for the quantum harmonic oscillator. We also present experiments uncovering a new adjoint-corrected approximation of the eigenvalues provided by the hybridizable discontinuous Galerkin method which converges with order 2k+2, as well as results showing the possibilities and limitations of using the adjoint-correction term as an asymptotically exact a posteriori error estimate.

A priori and a posteriori error estimates for a virtual element method for the non-self-adjoint Steklov eigenvalue problem

Abstract In this paper we analyze a virtual element method (VEM) for the non-self-adjoint Steklov eigenvalue problem. The conforming VEM on polytopal meshes is used for discretization. We analyze the correct spectral approximation of the discrete scheme and prove an a priori error estimate for the discrete eigenvalues and eigenfunctions. The convergence order of a discrete eigenvalue may decrease if the corresponding eigenfunction has a singularity and it can be improved on a locally refined mesh. The VEM has great flexibility in handling computational meshes. These facts motivate us to construct a computable a posteriori error estimator for the VEM and prove its reliability and efficiency. This estimator can be applied to very general polytopal meshes with hanging nodes. Finally, we show numerical examples to verify the theoretical results, including optimal convergence of discrete eigenvalues on uniformly refined meshes of a square domain and a cube domain, and we demonstrate the efficiency of the estimator on adaptively refined meshes on an L-shaped domain and also discuss the influence of stabilization parameters on the virtual element approximation.

Performance Analysis of FEM Solvers on Practical Electromagnetic Problems

The paper presents a comparative analysis of different commercial and academic software. The comparison aims to examine how the integrated adaptive grid refinement methodologies can deal with challenging, electromagnetic-field related problems. For this comparison, two bench-mark problems were examined in the paper. The first example is a solution of an L-shape domain like test problem, which has a singularity at a certain point in the geometry. The second problem is an induction heated aluminum rod, which accurate solution needs to solve non-linear, coupled physical fields. The accurate solution of this problem requires applying adaptive mesh generation strategies or applying a very fine mesh in the electromagnetic domain, which can significantly increase the computational complexity. The results show that the fully-hp adaptive meshing strategies, which are integrated into Agros Suite, can significantly reduce the task's computational complexity compared to the automatic h-adaptivity, which is part of the examined, popular commercial solvers.

A posteriori error estimates by weakly symmetric stress reconstruction for the Biot problem

A posteriori error estimates are constructed for the three-field variational formulation of the Biot problem involving the displacements, the total pressure and the fluid pressure. The discretization under focus is the H1(Ω)-conforming Taylor–Hood finite element combination, consisting of polynomial degrees k+1 for the displacements and the fluid pressure and k for the total pressure. An a posteriori error estimator is derived on the basis of H(div)-conforming reconstructions of the stress and flux approximations. The symmetry of the reconstructed stress is allowed to be satisfied only weakly. The reconstructions can be performed locally on a set of vertex patches and lead to a guaranteed upper bound for the error with a constant that depends only on local constants associated with the patches and thus on the shape regularity of the triangulation. Particular emphasis is given to nearly incompressible materials and the error estimates hold uniformly in the incompressible limit. Numerical results on the L-shaped domain confirm the theory and the suitable use of the error estimator in adaptive strategies.

A penalized weak Galerkin spectral element method for second order elliptic equations

A weak Galerkin spectral element method is proposed to solve second order partial differential equations. Following the idea of the weak Galerkin finite element method, this method introduces for the unknown solution a weak function consisting of one component on the elements together with one component on the element interfaces, and replaces derivatives in the standard variational form with weak derivatives defined on the space of weak functions on each element. As in a classic spectral element method, approximation spaces for weak functions on each triangular or parallelogram element are defined from orthogonal polynomials on the reference domain through a one-to-one mapping, and approximation spaces for weak derivatives are then established via the Piola transform after an insightful investigation of the weak gradient and the discrete weak gradient. To eliminate the effect of the possible nullity of the discrete weak gradient and guarantee the wellposedness of the resulting algebraic system, a penalty term defined on the edges is supplemented into the Galerkin approximation scheme. Error estimates for both the source problem and the eigenvalue problem on meshes consisting of affine families of triangles and quadrilaterals are obtained in the sequel, which are optimal in the mesh size and suboptimal by one-half order with respect to the polynomial degree. Numerical experiments for the eigenvalue problems are performed on both the typical square domain and L-shaped domain with triangular meshes and quadrilateral meshes, which illustrate the effectiveness and high accuracy of our penalized weak Galerkin spectral element method.

A Compact FEM Implementation for Parabolic Integro-Differential Equations in 2D

Although two-dimensional (2D) parabolic integro-differential equations (PIDEs) arise in many physical contexts, there is no generally available software that is able to solve them numerically. To remedy this situation, in this article, we provide a compact implementation for solving 2D PIDEs using the finite element method (FEM) on unstructured grids. Piecewise linear finite element spaces on triangles are used for the space discretization, whereas the time discretization is based on the backward-Euler and the Crank–Nicolson methods. The quadrature rules for discretizing the Volterra integral term are chosen so as to be consistent with the time-stepping schemes; a more efficient version of the implementation that uses a vectorization technique in the assembly process is also presented. The compactness of the approach is demonstrated using the software Matrix Laboratory (MATLAB). The efficiency is demonstrated via a numerical example on an L-shaped domain, for which a comparison is possible against the commercially available finite element software COMSOL Multiphysics. Moreover, further consideration indicates that COMSOL Multiphysics cannot be directly applied to 2D PIDEs containing more complex kernels in the Volterra integral term, whereas our method can. Consequently, the subroutines we present constitute a valuable open and validated resource for solving more general 2D PIDEs.