Standard Finite Element Method Research Articles

DeepXDE: A Deep Learning Library for Solving Differential Equations

Deep learning has achieved remarkable success in diverse applications; however, its use in solving partial differential equations (PDEs) has emerged only recently. Here, we present an overview of physics-informed neural networks (PINNs), which embed a PDE into the loss of the neural network using automatic differentiation. The PINN algorithm is simple, and it can be applied to different types of PDEs, including integro-differential equations, fractional PDEs, and stochastic PDEs. Moreover, from an implementation point of view, PINNs solve inverse problems as easily as forward problems. We propose a new residual-based adaptive refinement (RAR) method to improve the training efficiency of PINNs. For pedagogical reasons, we compare the PINN algorithm to a standard finite element method. We also present a Python library for PINNs, DeepXDE, which is designed to serve both as an educational tool to be used in the classroom as well as a research tool for solving problems in computational science and engineering. Specifically, DeepXDE can solve forward problems given initial and boundary conditions, as well as inverse problems given some extra measurements. DeepXDE supports complex-geometry domains based on the technique of constructive solid geometry and enables the user code to be compact, resembling closely the mathematical formulation. We introduce the usage of DeepXDE and its customizability, and we also demonstrate the capability of PINNs and the user-friendliness of DeepXDE for five different examples. More broadly, DeepXDE contributes to the more rapid development of the emerging scientific machine learning field.

Application of the generalized finite-element method to the acoustic wave simulation in exploration seismology

The generalized finite-element method (GFEM) has been applied frequently to solve harmonic wave equations, but its use in the simulation of transient wave propagation is still limited. We have applied GFEM for the simulation of the acoustic wave equation in models relevant to exploration seismology. We also perform an assessment of its accuracy and efficiency. The main advantage of GFEM is that it provides an enhanced solution accuracy compared to the standard finite-element method (FEM). This is attained by adding user-defined enrichment functions to standard FEM approximations. For the acoustic wave equation, we consider plane waves oriented in different directions as the enrichments, whose argument includes the largest wavenumber of the wavefield. We combine GFEM with an unconditionally stable time integration scheme with a constant time step. To assess the accuracy and efficiency of GFEM, we compare the GFEM simulation results against those obtained with the spectral-element method (SEM). We use SEM because it is the method of choice for wave propagation simulation due to its proven accuracy and efficiency. In the numerical examples, we first perform a convergence study in space and time, evaluating the accuracy of both methods against a semianalytical solution. Then, we consider simulations of relevant models in exploration seismology that include low-velocity features, an inclusion with a complex geometric boundary and topography. Results using these models indicate that GFEM presents comparable accuracy and efficiency to those based on SEM. For the given examples, the GFEM efficiency stems from the combined effect of local mesh refinement, nonconforming or unstructured, and the unconditionally stable time integration scheme with a constant time step. Moreover, these features provide great flexibility to the GFEM implementations, proving advantageous when using, for example, unstructured grids that impose severe time step size restrictions in SEM.

Implicit and semi-implicit second-order time stepping methods for the Richards equation

This study concerns numerical methods for efficiently solving the Richards equation where different weak formulations and computational techniques are analyzed. The spatial discretizations are based on standard or mixed finite element methods. Different implicit and semi-implicit temporal discretization techniques of second-order accuracy are studied. To obtain a linear system for the semi-implicit schemes, we propose second-order techniques using extrapolation formulas and/or semi-implicit Taylor approximations for the temporal discretization of nonlinear terms. A numerical convergence study and a series of numerical tests are performed to analyze efficiency and robustness of the different schemes. The developed scheme, based on the proposed temporal extrapolation techniques and the mixed formulation involving the saturation and pressure head and using the standard linear Lagrange element, performs better than other schemes based on the saturation and the flux and using the Raviart-Thomas elements. The proposed semi-implicit scheme is a good alternative when implicit schemes meet convergence issues.

Towards the simulation of active cardiac mechanics using a smoothed finite element method

In the last decades, various computational models have been developed to simulate cardiac electromechanics. The most common numerical tool is the finite element method (FEM). However, this method crucially depends on the mesh quality. For complex geometries such as cardiac structures, it is convenient to use tetrahedral discretisations which can be generated automatically. On the other hand, such automatic meshing with tetrahedrons together with large deformations often lead to elements distortion and volumetric locking. To overcome these difficulties, different smoothed finite element methods (S-FEMs) have been proposed in the recent years. They are known to be volumetric locking free, less sensitive to mesh distortion and so far have been used e.g. in simulation of passive cardiac mechanics. In this work, we extend for the first time node-based S-FEM (NS-FEM) towards active cardiac mechanics. Firstly, the sensitivity to mesh distortion is tested and compared to that of FEM. Secondly, an active contraction in circumferentially aligned fibre direction is modelled in the healthy and the infarcted case. We show, that the proposed method is more robust with respect to mesh distortion and computationally more efficient than standard FEM. Being furthermore free of volumetric locking problems makes S-FEM a promising alternative in modelling of active cardiac mechanics, respectively electromechanics.

Optimal strong convergence rates of some Euler-type timestepping schemes for the finite element discretization SPDEs driven by additive fractional Brownian motion and Poisson random measure

In this paper, we study the numerical approximation of a general second order semilinear stochastic partial differential equation (SPDE) driven by a additive fractional Brownian motion (fBm) with Hurst parameter H>frac {1}{2} and Poisson random measure. Such equations are more realistic in modelling real world phenomena. To the best of our knowledge, numerical schemes for such SPDE have been lacked in the scientific literature. The approximation is done with the standard finite element method in space and three Euler-type timestepping methods in time. More precisely the well-known linear implicit method, an exponential integrator and the exponential Rosenbrock scheme are used for time discretization. In contract to the current literature in the field, our linear operator is not necessary self-adjoint and we have achieved optimal strong convergence rates for SPDE driven by fBm and Poisson measure. The results examine how the convergence orders depend on the regularity of the noise and the initial data and reveal that the full discretization attains the optimal convergence rates of order mathcal {O}(h^{2}+varDelta t) for the exponential integrator and implicit schemes. Numerical experiments are provided to illustrate our theoretical results for the case of SPDE driven by the fBm noise.

An interface-enriched generalized finite element method for level set-based topology optimization

During design optimization, a smooth description of the geometry is important, especially for problems that are sensitive to the way interfaces are resolved, e.g., wave propagation or fluid-structure interaction. A level set description of the boundary, when combined with an enriched finite element formulation, offers a smoother description of the design than traditional density-based methods. However, existing enriched methods have drawbacks, including ill-conditioning and difficulties in prescribing essential boundary conditions. In this work, we introduce a new enriched topology optimization methodology that overcomes the aforementioned drawbacks; boundaries are resolved accurately by means of the Interface-enriched Generalized Finite Element Method (IGFEM), coupled to a level set function constructed by radial basis functions. The enriched method used in this new approach to topology optimization has the same level of accuracy in the analysis as the standard finite element method with matching meshes, but without the need for remeshing. We derive the analytical sensitivities and we discuss the behavior of the optimization process in detail. We establish that IGFEM-based level set topology optimization generates correct topologies for well-known compliance minimization problems.

Stability Analysis and Parameter Classification of a Reaction-Diffusion Model on an Annulus

This work explores the influence of domain-size on the evolution of pattern formation modelled by an activator-depleted reactiondiffusion system on a flat-ring (annulus). A closed form expression is derived for the spectrum of the Laplace operator on the domain. Spectral method is used to depict the close form solution on the domain. The bifurcation analysis of activator-depleted reactiondiffusion system is conducted on the admissible parameter space under the influence of domain-size. The admissible parameter space is partitioned under a set of proposed conditions relating the reactiondiffusion constants with the domain-size. Finally, the full system is numerically simulated on a two dimensional annular region using the standard Galerkin finite element method to verify the influence of the analytically derived domain-dependent conditions.

The double adaptivity paradigm: (How to circumvent the discrete inf–sup conditions of Babuška and Brezzi)

We present an efficient implementation of the double adaptivity algorithm of Cohen et al. (2012) within the setting of the Petrov–Galerkin method with optimal test functions. We apply this method to the ultraweak variational formulation of a general linear variational problem discretized with the standard Galerkin finite element method. As an example, we demonstrate the feasibility of the method in the context of the convection-dominated diffusion problem. The presented ideas, however, apply to virtually any well-posed system of first-order partial differential equations, including singular perturbation problems.

Weak convergence of fully discrete finite element approximations of semilinear hyperbolic SPDE with additive noise

The numerical approximation of the mild solution to a semilinear stochastic wave equation driven by additive noise is considered. A standard finite element method is employed for the spatial approximation and a a rational approximation of the exponential function for the temporal approximation. First, strong convergence of this approximation in both positive and negative order norms is proven. With the help of Malliavin calculus techniques this result is then used to deduce weak convergence rates for the class of twice continuously differentiable test functions with polynomially bounded derivatives. Under appropriate assumptions on the parameters of the equation, the weak rate is found to be essentially twice the strong rate. This extends earlier work by one of the authors to the semilinear setting. Numerical simulations illustrate the theoretical results.

Hyper-reduced arc-length algorithm for stability analysis in elastoplasticity

In this article an “hyper-reduced” scheme for the Crisfield’s algorithm (Crisfield, 1981) applied to buckling simulations and plastic instabilities is presented. The two linear systems and the ellipse equation entering the algorithm are projected on a reduced space and solved in a reduced integration domain, resulting in a system of “hyper-reduced” equations. Use is made of the Gappy proper orthogonal decomposition to recover stresses outside the reduced integration domain. Various methods are proposed to construct a reduced bases, making use of simulation data obtained with standard finite element method and a stress-based error criterion for the hyper reduced calculations is proposed. A “greedy” algorithm coupled with this error criterion is used to generate intelligently full standard finite element simulations and enrich the reduced base, demonstrating the adequacy of the error criterion. Finally, numerical results pertaining to elastoplastic structures undergoing finite strains, with emphasis on buckling and limit load predictions are presented. A parametric study on the geometry of the structure is carried out in order to determine the domain of validity of the proposed hyper-reduced modeling approach.

XFEM with partial Heaviside function enrichment for fracture analysis

In this work, an extended finite element method with partial Heaviside function enrichment (XFEM-P) in the crack tip element is proposed for fracture analysis for the first time. In this method, the crack tip element is divided into cracked and uncracked domains naturally by the crack geometry. In the cracked domain, Heaviside function enrichment is employed to simulate the crack discontinuity across the crack surface. In the uncracked domain, the displacement from standard finite element method is employed. Ramp function is used to ensure the displacement continuity in the method. Through this way, the crack tip can be arbitrarily located inside the crack tip element, which is a great advantage over traditional counterpart, i.e., XFEM with only Heaviside function enrichment (XFEM-H). Thus the mesh work for the XFEM-P is much easier than that of the XFEM-H, resulting in great conveniences for crack propagation. The new method is applied to several examples. From the computed results, it is observed that the XFEM-P offers more accurate results than the XFEM-H in terms of strain energy and SIFs. The XFEM-P performs better than XFEM-H in numerical properties such as convergence rate and condition number.

Hierarchical deep-learning neural networks: finite elements and beyond

The hierarchical deep-learning neural network (HiDeNN) is systematically developed through the construction of structured deep neural networks (DNNs) in a hierarchical manner, and a special case of HiDeNN for representing Finite Element Method (or HiDeNN-FEM in short) is established. In HiDeNN-FEM, weights and biases are functions of the nodal positions, hence the training process in HiDeNN-FEM includes the optimization of the nodal coordinates. This is the spirit of r-adaptivity, and it increases both the local and global accuracy of the interpolants. By fixing the number of hidden layers and increasing the number of neurons by training the DNNs, rh-adaptivity can be achieved, which leads to further improvement of the accuracy for the solutions. The generalization of rational functions is achieved by the development of three fundamental building blocks of constructing deep hierarchical neural networks. The three building blocks are linear functions, multiplication, and inversion. With these building blocks, the class of deep learning interpolation functions are demonstrated for interpolation theories such as Lagrange polynomials, NURBS, isogeometric, reproducing kernel particle method, and others. In HiDeNN-FEM, enrichment functions through the multiplication of neurons is equivalent to the enrichment in standard finite element methods, that is, generalized, extended, and partition of unity finite element methods. Numerical examples performed by HiDeNN-FEM exhibit reduced approximation error compared with the standard FEM. Finally, an outlook for the generalized HiDeNN to high-order continuity for multiple dimensions and topology optimizations are illustrated through the hierarchy of the proposed DNNs.

Enhanced Pseudo-Three-Dimensional Model for Multiple Hydraulic Fractures

Abstract Modeling the propagation of hydraulic fractures is a complex problem that involves solving the equations that describe the intimate coupling between the cracked and porous solid phase and the fracturing fluid that drives the process according to established crack extension criteria. Most of the existing computational techniques become prohibitively expensive if applied to three-dimensional configurations, especially those involving multiple fractures because their accuracy relies on extremely fine discretizations of the geometry. These include the standard finite element method and standard displacement discontinuity method, which require substantial mesh densities in the vicinity of the crack front to capture the complex nature of the singular fields associated with fluid-driven cracks. Models have recently been proposed that offer sufficient accuracy and a significant increase in computational efficiency. These include the enhanced pseudo-3D model that explicitly incorporates a non-local elasticity relation and known crack-front asymptotic solutions that capture the multiscale near-front behavior. Here, we present an enhanced pseudo-3D model for simulating the propagation of multiple planar and nonplanar hydraulic fractures in a linear elastic medium. The algorithm is built on a fixed mesh with an adaptive tip element, and it incorporates leak-off, as well as stress interaction between the fractures. The model is validated against reference solutions of plane strain, radial, and multi-planar fracture geometries, for various sets of parameters that cover the majority of the problem parametric space.

Strong convergence of a linearization method for semi-linear elliptic equations with variable scaled production

This work is devoted to the development and analysis of a linearization algorithm for microscopic elliptic equations, with scaled degenerate production, posed in a perforated medium and constrained by the homogeneous Neumann–Dirichlet boundary conditions. This technique plays two roles: to guarantee the unique weak solvability of the microscopic problem and to provide a fine approximation in the macroscopic setting. The scheme systematically relies on the choice of a stabilization parameter in such a way as to guarantee the strong convergence in $$H^1$$ norm for both the microscopic and macroscopic problems. In the standard variational setting, we prove the $$H^1$$ -type contraction at the micro-scale based on the energy method. Meanwhile, we adopt the classical homogenization result in line with corrector estimate to show the convergence of the scheme at the macro-scale. In the numerical section, we use the standard finite element method to assess the efficiency and convergence of our proposed algorithm.

Numerical Analysis of Microcoil‐Induced Electric Fields and Evaluation of In vivo Magnetic Stimulation of the Mouse Brain

Magnetic stimulation is widely used as a medical treatment for neurological diseases. A reduction in the size of magnetic stimulation devices is desirable, however, because the coils are relatively large compared with electrode stimulation devices. Furthermore, in transcranial magnetic stimulation, the exact magnetically evoked areas in the brain and the mechanisms of neural activation are largely unknown. This study aimed to develop a new implantable microcoil device that stimulates the brain locally, on the order of several tens of micrometers, as well as to understand the mechanisms of micromagnetic stimulation‐induced neural activity. First, to investigate the effects of microcoil shape on neural activation, the induced electric fields of semicircular microcoils with different diameters were calculated numerically in a microcoil model using the standard finite element method. Next, on the basis of the obtained numerical results from the microcoil model with different diameters, we compared the spatial properties of the possible activated areas in neural tissue and examined the most effective microcoil shape for neural activation. Finally, we measured microcoil‐evoked responses using autofluorescent flavoprotein imaging of the mouse brain in vivo, evaluated the validity of our proposed microcoil devices, and discussed possible future improvements for a chronic implant. © 2020 Institute of Electrical Engineers of Japan. Published by Wiley Periodicals LLC.

Analysis of transient wave propagation in inhomogeneous media using edge-based gradient smoothing technique and bathe time integration method

As is known to all, the classical finite element method (FEM) always fails to effectively solve the wave problems in the relatively large wave numbers due to the troublesome numerical error issue. In this paper, the standard FEM with edge-based gradient smoothing technique and Bathe time integration method is developed to analyze the transient wave propagation problems in inhomogeneous media. We explicitly show that the numerical error of the calculated solutions for transient wave propagation problems consists of two different parts, namely the spatial discretization error and the temporal discretization error. Due to the edge-based gradient smoothing technique and the appropriate numerical dissipation effects from Bathe time integration scheme, it is found that the total numerical error can be significantly suppressed and more accurate numerical solutions can be obtained. Several typical numerical examples have been conducted to examine the capacity of the proposed method in solving transient wave propagation problems in inhomogeneous media.

Numerical solution of Rosseland model for transient thermal radiation in non-grey optically thick media using enriched basis functions

Heat radiation in optically thick non-grey media can be well approximated with the Rosseland model which is a class of nonlinear diffusion equations with convective boundary conditions. The optical spectrum is divided into a set of finite bands with constant absorption coefficients but with variable Planckian diffusion coefficients. This simplification reduces the computational costs significantly compared to solving a full radiative heat transfer model. Therefore, the model is very popular for industrial and engineering applications. However, the opaque nature of the media often results in thermal boundary layers that requires highly refined meshes, to be recovered numerically. Such meshes can significantly hinder the performance of numerical methods. In this work we explore for the first time using enriched basis functions for the model in order to avoid using refined meshes. In particular, we discuss the finite element method when using basis functions enriched with a combination of exponential and hyperbolic functions. We show that the enrichment can resolve thermal boundary layers on coarse meshes and with few elements. Comparisons to the standard finite element method for thermal radiation in non-grey optically thick media with multi-frequency bands show the efficiency of the approach. Although we mainly study the enriched basis functions in glass cooling applications the substantial saving in the computational requirements makes the approach highly relevant to a large number of engineering applications that involve solving the Rosseland model.

Member Sizing Optimization of Large Scale Steel Space Trusses Using a Symbiotic Organisms Search Algorithm

A systematic approach of optimization is needed to achieve an optimal design of large and complex truss structures. In the last three decades, several researchers have developed and applied various metaheuristic optimization methods to the design of truss structures. This paper investigates a new metaheuristic algorithm called symbiotic organisms search (SOS) for member sizing optimization of relatively large steel trusses. The case studies include a 120-bar dome truss and a 942-bar tower truss. The structural analyses are carried out using the standard finite element method. The profiles of the truss members are circular hollow structural sections selected from a set of the American Institute of Steel Construction standard profiles. The design results using the SOS are then compared to those obtained using other metaheuristic methods, namely the particle swarm optimization, differential evolution, and teaching-learning-based optimization. The comparison shows the superior performance of the SOS in terms of the optimal solution, consistency, and convergence. Thus, the SOS is a good alternative for optimizing the design of steel truss structures in real engineering practice.

A unified analysis of a class of quadratic finite volume element schemes on triangular meshes

This paper presents a general framework for the coercivity analysis of a class of quadratic finite volume element (FVE) schemes on triangular meshes for solving elliptic boundary value problems. This class of schemes covers all the existing quadratic schemes of Lagrange type. With the help of a new mapping from the trial function space to the test function space, we find that each element matrix can be decomposed into three parts: the first part is the element stiffness matrix of the standard quadratic finite element method (FEM), the second part is the difference between the FVE and FEM on the element boundary, while the third part can be expressed as the tensor product of two vectors. Thanks to this decomposition, we obtain a sufficient condition to guarantee the existence, uniqueness, and coercivity result of the FVE solution on triangular meshes. Moreover, based on this sufficient condition, some minimum angle conditions with simple, analytic, and computable expressions can be derived and they depend only on the constructive parameters of the schemes. As a byproduct, some existing coercivity results are improved. Finally, an optimal H1 error estimate is proved by the standard techniques.

A topology optimisation of composite elastic metamaterial slabs based on the manipulation of far-field behaviours

We propose a numerical method for a topology optimisation of composite elastic metamaterial slabs. We aim to realise some anomalous functionalities such as perfect absorption, wave-mode conversion, and negative refraction by designing the shape and topology of (visco-)elastic inclusions. Instead of manipulating effective material constants, we propose to utilise the far-field characteristics of scattered waves. This allows us to achieve novel functionalities for waves in not only low- but also high-frequency ranges. The design sensitivity corresponding to the far-field characteristics is rigorously derived using the adjoint variable method and incorporated into a level-set–based topology optimisation algorithm. The design sensitivity is computed by the boundary element method with periodic Green’s function instead of the standard finite element method to rigorously deal with the radiation of scattered waves without absorbing boundaries. We show some numerical examples to demonstrate the effectiveness of the proposed method.