Domain Boundary Conditions Research Articles

Neural network-augmented differentiable finite element method for boundary value problems

Classical numerical methods such as finite element method (FEM) face limitations due to their low efficiency when addressing large-scale problems. As a novel paradigm, the physics-informed neural network (PINN) has demonstrated significant potential to solve partial differential equations. However, conventional PINNs utilize meshless control at discrete sampling points, which limits their ability to effectively handle complex boundaries. Moreover, catastrophic failure may occur in the deep energy method (DEM, a specific type of PINN). To handle these challenges, this study proposes a Neural Network-augmented Differentiable Finite Element Method (NNDFEM) by combining PINN and finite element approximation. In NNDFEM, the neural network backend solely predicts nodal variables. Derivatives and complex boundary conditions can be well handled by the finite element frontend. The governing equation over the domain, Dirichlet, and Neumann boundary conditions are directly enforced on the finite element frontend. Thus, losses of boundary conditions in PINN are rendered unnecessary. The overfitting problem in DEM is also significantly mitigated. Fully connected neural network (FCNN), modified FCNN, and graph-convolutional network are tested as backends. NNDFEM circumvents nodal force calculation and matrix assembly in FEM. Functional losses of linear elasticity, finite strain nonlinear elasticity, heat conduction, and flow in porous media are validated. A systematic exploration unveils the role of 3D finite element mesh. For large-scale problems, a multi-fidelity learning strategy is employed. Thus, the three-dimensional case with over three million degrees of freedom trains well in two minutes. Benefiting from the fast inference of the neural network backend, the forward pass is 8,550 times faster than FEM.

Homogenization and corrector results of elliptic problems with Signorini boundary conditions in perforated domains

Homogenization and corrector results of elliptic problems with Signorini boundary conditions in perforated domains

Analysis of efficient preconditioner for solving Poisson equation with Dirichlet boundary condition in irregular three-dimensional domains

This paper analyzes modified ILU (MILU)-type preconditioners for efficiently solving the Poisson equation with Dirichlet boundary conditions on irregular domains. In [1], the second-order accuracy of a finite difference scheme developed by Gibou et al. [2] and the effect of the MILU preconditioner were presented for two-dimensional problems. However, the analyses do not directly extend to three-dimensional problems. In this paper, we first demonstrate that the Gibou method attains second-order convergence for three-dimensional irregular domains, yet the discretized Laplacian exhibits a condition number of O(h−2) for a grid size h. We show that the MILU preconditioner reduces the order of the condition number to O(h−1) in three dimensions. Furthermore, we propose a novel sectored-MILU preconditioner, defined by a sectorized lexicographic ordering along each axis of the domain. We demonstrate that this preconditioner reaches a condition number of order O(h−1) as well. Sectored-MILU not only achieves a similar or better condition number than conventional MILU but also improves parallel computing efficiency, enabling very efficient calculation when the dimensionality or problem size increases significantly. Our findings extend the feasibility of solving large-scale problems across a range of scientific and engineering disciplines.

On Operator Estimates for Elliptic Operators with Mixed Boundary Conditions in Two-Dimensional Domains with Fast Oscillating Boundary

On Operator Estimates for Elliptic Operators with Mixed Boundary Conditions in Two-Dimensional Domains with Fast Oscillating Boundary

Homogenization of quasilinear problems with semilinear terms and Signorini boundary conditions in perforated domains

Homogenization of quasilinear problems with semilinear terms and Signorini boundary conditions in perforated domains

Construction and investigation of differential hierarchical models for the Newtonian fluids

AbstractWe construct mathematical hierarchical models in Eulerian coordinates for an ideal and a Newtonian viscous fluid flow in the prismatic shell‐like domains. The peculiarities of well‐posedness of boundary conditions for angular domains are discussed.

Nonlinear elliptic eigenvalue problems in cylindrical domains becoming unbounded in one direction

The aim of this work is to characterize the asymptotic behaviour of the first eigenfunction of the generalised p-Laplace operator with mixed (Dirichlet and Neumann) boundary conditions in cylindrical domains when the length of the cylindrical domains tends to infinity. This generalises an earlier work of Chipot et al. (Asymptot. Anal. 85(3–4) (2013) 199–227) where the linear case p = 2 is studied. Asymptotic behavior of all the higher eigenvalues of the linear case and the second eigenvalues of general case (using topological degree) for such problems is also studied.

General energy decay estimates for a variable coefficient viscoelastic semilinear wave equation with logarithmic source term and acoustic boundary conditions

In this paper, we consider a variable coefficient viscoelastic semilinear wave equation with logarithmic source term and acoustic boundary conditions in domains with non‐locally reacting boundary. By potential well method, we get the global existence of solution and through several important lemmas, we prove the general energy decay estimate of the system.

Defeaturing of CAD Models Using Machine Learning

Part features, such as dents, holes, and bumps, are integral to the thin-walled sheet metal components used in the automobile and aerospace industries. These features are often required to be suppressed for more agile and reasonably accurate results during the CAE analysis. However, identifying these features is a critical task usually performed by experts. It requires a detailed analysis of the CAD model, which largely depends on the importance of each feature, feature size, and domain boundary conditions. Thus, identifying such features is a challenging task. This work proposes a novel data-driven approach to create an intuitive model that identifies suppressible features on CAD models of sheet metal parts. The dataset to train the supervised learning model is generated by extensive finite element analysis of the part models. The case studies show that the trained Machine learning model gave good test accuracy.

Some effects of domain size and boundary conditions on the accuracy of airfoil simulations

This paper investigates a specific case of one of the most popular fluid dynamic simulations, the incompressible flow around an airfoil (NACA 0012 here) at a high Reynolds number (6×106\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$6 \ imes 10^6$$\\end{document}). OpenFOAM software was used to study the effect of domain size and four common choices of boundary conditions on airfoil lift, drag, surface friction, and pressure. We also examine the relation between boundary conditions and the velocity, pressure, and vorticity distributions throughout the domain. In addition to the common boundary conditions, we implement the “point vortex” boundary condition that was introduced many years ago but is now rarely used. We also applied the point vortex condition for the outlet pressure instead of using the traditional Neumann condition. With the airfoil generating significant lift at incidence angles of 5∘,10∘\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$5^\\circ , 10^\\circ$$\\end{document}, and 14∘\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$14^\\circ$$\\end{document}, we confirm a previous finding that the boundary conditions combine with domain size to produce an induced (pressure) drag. The change in the pressure drag with domain size is significant for the commonly-used boundary conditions but is much smaller for the point vortex alternative. The point vortex boundary condition increases the execution time, but this is more than offset by the reduction in domain size needed to achieve a specified accuracy in the lift and drag. This study also estimates the error in total drag and lift due to domain size and shows it can be almost eliminated using the point vortex boundary condition. We also used the impulse form of the momentum equations to study the relation between drag and lift and spurious vorticity, which is generated as a result of using non-exact boundary conditions. These equations reveal that the spurious vorticity throughout the domain is associated with cancelling circulation around the domain boundaries.

Learning domain-independent Green’s function for elliptic partial differential equations

Green’s function characterizes a partial differential equation (PDE) and maps its solution in the entire domain as integrals. Finding the analytical form of Green’s function is a non-trivial exercise, especially for a PDE defined on a complex domain or a PDE with variable coefficients. In this paper, we propose a novel boundary integral network to learn the domain-independent Green’s function, referred to as BIN-G. We evaluate the Green’s function in the BIN-G using a radial basis function (RBF) kernel-based neural network. We train the BIN-G by minimizing the residual of the PDE and the mean squared errors of the solutions to the boundary integral equations for prescribed test functions. By leveraging the symmetry of the Green’s function and controlling refinements of the RBF kernel near the singularity of the Green function, we demonstrate that our numerical scheme enables fast training and accurate evaluation of the Green’s function for PDEs with variable coefficients. The learned Green’s function is independent of the domain geometries, forcing terms, and boundary conditions in the boundary integral formulation. Numerical experiments verify the desired properties of the method and the expected accuracy for the two-dimensional Poisson and Helmholtz equations with variable coefficients.

A stable BIE method for the Laplace equation with Neumann boundary conditions in domains with piecewise smooth boundaries

A stable BIE method for the Laplace equation with Neumann boundary conditions in domains with piecewise smooth boundaries

An improved plate deep energy method for the bending, buckling and free vibration problems of irregular Kirchhoff plates

An improved plate deep energy method without penalty terms is introduced to study the bending, free vibration and buckling problems of irregular Kirchhoff plates. The finite difference produced by convolution operation is employed for a faster calculation of derivatives and enforcement of boundary conditions. Most deep learning based neural network methods adopt the penalty method to impose boundary constraints, which may cause convergence issues especially when dealing with complex solution domain or mixed boundary conditions. In this paper, the R-function is utilized to build the distance function multiplied by the neural network output to meet the Dirichlet boundary constraints, while the boundary constraints expressed by derivative functions are applied by introducing virtual nodes. With such treatments, the boundary losses of a plate system can be avoided. To validate the proposed method, numerical examples considering plates with irregular shapes and mixed boundary conditions are studied, and the results are compared to those provided by the theoretical solutions, finite element method (FEM), and literature.

Modeling Dynamics of Laterally Impacted Piles in Gravel Using Erosion Method

Understanding the dynamic interaction between piles and the surrounding soil under vehicular impacts is essential for effectively designing and optimizing soil-embedded vehicle barrier systems. The complex behavior of pile–soil systems under impact loading, attributed to the soil’s nonlinear behavior and large deformation experienced by both components, presents significant simulation challenges. Popular computation techniques, such as the updated Lagrangian finite element method (UL-FEM), encounter difficulties in scenarios marked by large soil deformation, e.g., impacts involving rigid piles. While mesh-free particle and discrete element methods offer another option, their computational demands for field-scale pile–soil impact simulations are considerable. We introduce the erosion method to bridge this gap by integrating UL-FEM with an erosion algorithm for simulating large soil deformations during vehicular impacts. Validation against established physical impact tests confirmed the method’s effectiveness for flexible and rigid pile failure mechanisms. Additionally, this method was used to examine the effects of soil mesh density, soil domain sizes, and boundary conditions on the dynamic impact response of pile–soil systems. Our findings provide guidelines for optimal soil domain size, mesh density, and boundary conditions. This investigation sets the stage for improved, computationally efficient techniques for the pile–soil impact problem, leading to better pile designs for vehicular impacts.

IF-TONIR: Iteration-free Topology Optimization based on Implicit Neural Representations

Topology optimization holds great significance as a research topic in the field of mechanical engineering, aiming to design and optimize structures to achieve desired performance while adhering to specific constraints. However, its high computational complexity and iterative optimization process severely impact the efficiency, which presents substantial obstacles to its practical applications. To tackle this challenge, recent research is dedicated to the advancement of iteration-free topology optimization methods that leverage neural networks and deep learning, aiming to directly predict optimal structures through optimization problem configurations. In this paper, we propose IF-TONIR, a novel data-driven topology optimization method that utilizes implicit neural representations. Our approach employs signed distance fields to represent structures, offering compact and smooth representations that effectively eliminate the checkerboard phenomenon commonly observed in density-based methods. IF-TONIR leverages Conditional Variational Autoencoders, which use a CNN-based encoder and a MLP-based decoder to learn and reconstruct optimal structures. We employ the features extracted from physical information as conditions to guide the decoder in generating optimal structures that adhere to specific design domain shapes and boundary conditions. Furthermore, we propose the integration of a topological loss based on persistent homology to train the model. This loss function effectively penalizes the existence of structural disconnections in the reconstructed output, thereby enhancing the overall physical reliability of the generated structures. Various experiments have demonstrated that our iteration-free topology optimization method based on implicit representations can accurately identify regions of high strain energy and generate continuous structures with low compliance. The methods also holds the theoretical capability of outputting optimal structures at any desired resolution. Our code and dataset are available on https://github.com/jbHu67/IF-TONIR.git

Quasilinear problems with nonlinear boundary conditions in higher-dimensional thin domains with corrugated boundaries

AbstractIn this work, we analyze the asymptotic behavior of a class of quasilinear elliptic equations defined in oscillating(N+1)\left(N+1)-dimensional thin domains (i.e., a family of bounded open sets fromRN+1{{\mathbb{R}}}^{N+1}, with corrugated bounder, which degenerates to an open bounded set inRN{{\mathbb{R}}}^{N}). We also allow monotone nonlinear boundary conditions on the rough border whose magnitude depends on the squeezing of the domain. According to the intensity of the roughness and a reaction coefficient term on the nonlinear boundary condition, we obtain different regimes establishing effective homogenized limits inNN-dimensional open bounded sets. In order to do that, we combine monotone operator analysis techniques and the unfolding method used to deal with asymptotic analysis and homogenization problems.

An optimal-transport finite-particle method for mass diffusion

We formulate a class of velocity-free finite-particle methods for mass transport problems based on a time-discrete incremental variational principle that combines entropy and the cost of particle transport, as measured by the Wasserstein metric. The incremental functional is further spatially discretized into finite particles, i.e., particles characterized by a fixed spatial profile of finite width, each carrying a fixed amount of mass. The motion of the particles is then governed by a competition between the cost of transport, that aims to keep the particles fixed, and entropy maximization, that aims to spread the particles so as to increase the entropy of the system. We show how the optimal width of the particles can be determined variationally by minimization of the governing incremental functional. Using this variational principle, we derive optimal scaling relations between the width of the particles, their number and the size of the domain. We also address matters of implementation, including the acceleration of the computation of diffusive forces by exploiting the Gaussian decay of the particle profiles and by instituting fast nearest-neighbor searches. Sources, advection, boundary flux and diffusion are accounted for by fractional steps We demonstrate the robustness and versatility of the finite-particle method by means of a test problem concerned with the injection of mass into a sphere. There test results demonstrate the meshless character of the method in any spatial dimension, its ability to redistribute mass particles and follow their evolution in time, its ability to satisfy flux boundary conditions for general domains based solely on a distance function, and its robust convergence characteristics.

Length and Boundary Condition Effects in the Buckling of Cylinders Under Different Load Combinations

AbstractThe newest version of shells buckling standard prEN 1993‐1‐6 (2021) adopts linear elastic buckling stress expressions for steel perfect cylinders under axial, circumferential, and torsional effects separately. However, there are no explicit detailed rules that cover the influence of two or more loading combinations on the cylinder buckling resistance, considering a full‐length domain and different boundary conditions. Consequently, this paper looks into such a topic, which will be a useful reference for future research extensions. The examined cylinders were subjected to three loading configurations: axial plus pressure, axial plus torsion, and axial, pressure, and torsion altogether. The finite element (FE) buckling capacities were compared with the prEN 1993‐1‐6 (2021) buckling strengths, from which several findings were obtained. Aspects such as the load configuration applied to a cylinder, the length domain in which a cylinder falls, and the boundary type of a cylinder's ends each led to distinctive and complex descriptions of buckling behavior. The outcomes of this study will in turn assist in developing the current standard prEN 1993‐1‐6 (2021) and its recommendation ECCS EDR5 (2013), particularly the chapter that is relevant to cylindrical shells of constant wall thickness under general loading.

Finite Difference Method for Solving the Two-Dimensional Laplace Equation in Curvilinear Coordinates

Abstract This paper presents a numerical method for solving the two-dimensional Laplace equation in curvilinear coordinates using a finite difference scheme. Laplace equation is a differential equation used to describe various physical phenomena studied within different branches of engineering. The two-dimensional Laplace equation has analytic solutions only for simple shaped domains and simple boundary conditions (first and second kind). Even so, the results are usually complicated to interpret and use. The present paper proposes a method to solve the equation for domains bounded by complex inner and outer curves. The boundary curves are defined by a set of points from which the equations of curves are approximated by using Fourier series. The domain is generated by blending the two curves using a blending function. The results are a set of continuous parametric equations which are defining the domain in angle and radius variables. Following a method presented in the literature, the Laplace equation is transformed from Cartesian coordinates to curvilinear coordinates, as the Fourier series is easily differentiable. Next, the new Laplace equation is approximated using the finite difference method and a MATLAB code is developed to solve the system of linear equations using successive over-relaxation method (SOR). The presented method is used to simulate steady two-dimensional heat conduction in a polygonal shape domain, with uniform and non-uniform Dirichlet boundary conditions. Results are analyzed in order to determine the model’s precision, speed and convergence criteria.

A Practical Walk-on-Boundary Method for Boundary Value Problems

We introduce the walk-on-boundary (WoB) method for solving boundary value problems to computer graphics. WoB is a grid-free Monte Carlo solver for certain classes of second order partial differential equations. A similar Monte Carlo solver, the walk-on-spheres (WoS) method, has been recently popularized in computer graphics due to its advantages over traditional spatial discretization-based alternatives. We show that WoB's intrinsic properties yield further advantages beyond those of WoS. Unlike WoS, WoB naturally supports various boundary conditions (Dirichlet, Neumann, Robin, and mixed) for both interior and exterior domains. WoB builds upon boundary integral formulations, and it is mathematically more similar to light transport simulation in rendering than the random walk formulation of WoS. This similarity between WoB and rendering allows us to implement WoB on top of Monte Carlo ray tracing, and to incorporate advanced rendering techniques (e.g., bidirectional estimators with multiple importance sampling, the virtual point lights method, and Markov chain Monte Carlo) into WoB. WoB does not suffer from the intrinsic bias of WoS near the boundary and can estimate solutions precisely on the boundary. Our numerical results highlight the advantages of WoB over WoS as an attractive alternative to solve boundary value problems based on Monte Carlo.