Standard Finite Element Method Research Articles

Meshfree multiscale method for the infiltration problem in permafrost

This study introduces a multiscale simulation approach, employing the Meshfree Generalized Multiscale Finite Element Method (GMsFEM), to numerically model fluid seepage in permafrost soil conditions. Combining Meshfree and Finite Element Methods, the coarse-grid system is addressed, while the standard Finite Element Method operates on the fine grid. For representing fractures at the fine grid level, the Discrete Fracture Model (DFM) is utilized. The findings demonstrate the method’s efficacy in accurately depicting soil heterogeneity and the impact of fractures on soil temperature.

A linearly implicit finite element full-discretization scheme for SPDEs with nonglobally Lipschitz coefficients

Abstract The present article deals with strong approximations of additive noise driven stochastic partial differential equations (SPDEs) with nonglobally Lipschitz nonlinearity in a bounded domain $ \mathcal{D} \in{\mathbb{R}}^{d}$, $ d \leq 3$. As the first contribution, we establish the well-posedness and regularity of the considered SPDEs in space dimension $d \le 3$, under more relaxed assumptions on the stochastic convolution. This improves relevant results in the literature and covers both the space-time white noise ($d=1$) and the trace-class noises ($\text{Tr} (Q) < \infty $) in multiple dimensions $d=2,3$. Such an improvement is achieved based on a key perturbation estimate for a perturbed PDE, with the aid of which we prove the convergence and uniform regularity of a spectral approximation of the SPDEs and thus get the improved regularity results. The second contribution of the paper is to propose and analyze a spatio-temporal discretization of the SPDEs, by incorporating a standard finite element method in space and a linearly implicit nonlinearity-tamed Euler method for the temporal discretization. The proposed time-stepping scheme is linearly implicit and does not suffer from solving nonlinear algebra equations as the backward Euler scheme does. Based on the improved regularity results, we recover the expected strong convergence rates of the fully discrete scheme and reveal how the convergence rates rely on the regularity of the noise process. In particular, a classical convergence rate of order $O(h^{2} +\tau )$ can be obtained even in high dimension $d=3$, as the driven noise is of trace class and satisfies certain regularity assumptions. The optimal error estimates turn out to be challenging and face some essential difficulties when the tamed time-stepping scheme meets the finite element spatial discretization, particularly in the context of low regularity and multiple dimensions $d \le 3$. Some highly nontrivial arguments are introduced to overcome the difficulties. Finally, numerical examples corroborate the claimed strong orders of convergence.

A nonlinear finite volume element method preserving the discrete maximum principle for heterogeneous anisotropic diffusion equations

In this paper, we propose a new nonlinear conservative and maximum-principle-preserving finite volume element method for heterogeneous anisotropic diffusion equations on triangular or strictly convex quadrilateral meshes. First, we decompose the numerical flux of the standard finite volume element method on each dual edge into two parts: main part having a two-point-flux structure, and residual part owning a multi-point-flux structure. Then, according to the sign of the residual part and the distribution of the local extremum, the residual part is modified to a two-point-flux structure with a nonlinear term. The resulting nonlinear scheme not only inherits conservation and accuracy of original scheme, but also possesses a maximum-preserving structure without extra restrictions on the meshes and diffusion coefficients. Several numerical experiments verify that our nonlinear scheme has an optimal convergence order and satisfies the discrete maximum principle.

SUPG-based stabilized finite element computations of convection-dominated 3D elliptic PDEs using shock-capturing

This study is interested in stabilized finite element computations of advection-dominated convection–diffusion-type partial differential equations. The governing equations are assumed to be defined on the 3D unit cube and stationary. Towards that end, we stabilize the standard (Bubnov–) Galerkin finite element method (GFEM), which typically suffers from yielding numerical approximations polluted with node-to-node nonphysical oscillations in convection dominance, by employing the streamline-upwind/Petrov–Galerkin (SUPG) formulation. In order to achieve better shock representations near sharp layers, the SUPG-stabilized formulation is also enhanced with a simple and residual-based shock-capturing mechanism, the so-called YZβ technique. Several test computations are performed to assess the efficiency and robustness of the proposed formulation. The test problems are examined under more challenging conditions than those previously studied in the literature, i.e., for flows substantially dominated by convection phenomena. The numerical results and comparisons demonstrate that the SUPG-YZβ combination significantly eliminates both globally spread and localized numerical instabilities. Beyond that, the proposed formulation accomplishes that by using only linear elements on relatively coarser meshes and without making use of any adaptive mesh strategies.

Size Optimization of Truss Structures Based on Genetic Algorithm by Developing Single-point Crossover with Bit-wise Mutation Using FORTRAN Programming

A truss is a collection of axially loaded structural elements connected through pin joints. Optimization of truss structures indicates the determination of best possible conditions that are necessary for achieving the most economical design in terms of lowest possible weights of the truss elements. According to recent studies, Size, Topology and Shape based optimizations are the three possible independent optimization scopes for finding the optimum weight of a truss. The current study focuses on the size optimization of truss structure and aims to identify the most cost-effective sections of the truss using a powerful evolutionary optimization technique named as Genetic Algorithm (GA). Here single point cross over method and bit wise mutation operator are adopted for expanding the search space with the assigned displacement and stress constraints. Fortran based sub routines are developed for each individual steps of GA such as fitness evaluation, selection, cross-over and mutation. Finaly the complete program is run for validation of its outcome. While comparing with the solution of an existing literature regarding a minimization of objective function having two design variables the outcomes tend to show a good correlation. Further, the study is extended for the weight minimization of two widely used truss having 10 members and 17 members respectively with standard Finite Element Method of analysis. Minimum weight of each truss is obtained via convergence plot and compared with the existing literatures. With this simple but efficient global optimization technique optimum weight of truss structure can easily be achieved.

A Timoshenko-Ehrenfest beam model for simulating Langevin transducer dynamics

The Langevin transducer is a widely used power ultrasonic device across both medical and industrial applications, from orthopaedic surgery to drilling and welding. It is a sandwich-type device which typically consists of piezoelectric ceramic rings between two metallic end-masses. The transducer is commonly operated at a particular resonant mode to deliver ultrasonic vibrations to a target structure of interest, and is generally modelled using approaches including the transfer matrix method and electromechanical equivalent circuit methods. Here, we propose a variational framework to study the dynamics of the Langevin transducer based on the Timoshenko-Ehrenfest beam theory and Hamilton's principle. The variational equation derived from this model is then discretized by the standard finite element method with spectral elements. To verify the proposed one-dimensional electromechanical model, the computed resonance frequencies, or natural frequencies, of the one-dimensional model are compared to those of a three-dimensional finite element model with respect to varying geometry parameters characterizing the transducer. The results of the reduced one-dimensional and full three-dimensional models are then compared to those measured through an experimental investigation consisting of laser Doppler vibrometry. This is undertaken for the first ten eigenfrequencies, including longitudinal and bending modes, where normalized amplitudes and vibration mode shapes are reported. The close correlation between modelling and experiment demonstrates that the proposed one-dimensional electromechanical model can deliver results consistent with the full three-dimensional model and from experiment, thus verifying a rapid and reliable method for studying the free vibrations and dynamics of the Langevin transducer which accounts for the axial vibrations of the piezoelectric ceramic stack in all cases.

Fast Two-Grid Finite Element Algorithm for a Fractional Klein- Gordon Equation

In this article, we propose a spatial two-grid finite element algorithm combined with a shifted convolution quadrature (SCQ) formula for solving the fractional Klein-Gordon equation. The time direction at tn − θ is approximated utilizing a second-order SCQ formula, where θ is an arbitrary constant. The spatial discretization is performed using a two-grid finite element method involving three steps: calculating the numerical solution by solving a nonlinear system iteratively on the coarse grid, obtaining the interpolation solution based on the computed solutions in the first step, and solving a linear finite element system on the fine grid. We present a numerical algorithm, validate the two-grid finite element method’s effectiveness, and demonstrate the computational efficiency for our method by the comparison of the computing results between the two-grid finite element method and the standard finite element method.

Geotechnical analysis involving strain localization of overconsolidated soils based on unified hardening model with hardening variable updated by a composite scheme

AbstractStrain localization simulation of overconsolidated soils with high overconsolidation ratio (OCR) has been a long‐standing challenge. Some critical state soil models, including the modified Cam‐clay (MCC) model, have been widely applied, but they may not predict the shear dilatancy of overconsolidated soils well in some cases. Hence, the unified hardening (UH) model, which may be viewed as a generalized version of the MCC model, is implemented. It has been recognized, nonetheless, that without resorting to the regularization mechanism, the standard finite element method (FEM) or the second‐order cone programming optimized finite element method (FEM‐SOCP) often experiences instability or interruption of calculating the hardening‐softening responses of overconsolidated soils. To resolve the aforementioned difficulty, the UH model is developed and implemented in the framework of FEM‐SOCP based on the micropolar continuum (mpcFEM‐SOCP) to predict strain localizations of overconsolidated soils. Furthermore, to obviate non‐convexity of mpcFEM‐SOCP induced by material softening, an effective composite update scheme of hardening variable pc, which refers to the implicit variable (IV) scheme for the hardening stage and then refers to the explicit variable (EV) scheme for the softening stage, is proposed. Based on one biaxial compression problem and one rigid strip footing problem, numerical analyses disclose that by applying mpcFEM‐SOCP in conjunction with the composite update scheme of pc, the UH model of micropolar continuum can effectively predict the strain localization behavior of overconsolidated soil during its failure stage, and the stable hardening‐softening responses of overconsolidated soils can be readily attained.

Superconvergence analysis of a two-grid BDF2-FEM for nonlinear dispersive wave equation

The aim of this paper is to study the superconvergent behavior of a two-grid finite element method (FEM) with 2-step backward differential formula (BDF2) for nonlinear dispersive wave equation. By introducing an auxiliary variable q=ut for the original variable u, the problem is transformed into a parabolic system. With the help of the combination technique of the interpolation and Ritz projection, the superclose and superconvergent estimates for the above two variables in H1-norm are obtained under the lower regularity of the exact solution compared with the standard Galerkin FEM. Finally, some numerical results are provided to show the correctness of the theoretical predictions and good performance of the proposed method.

Directional enrichment functions for finite element solutions of transient anisotropic diffusion

The present study proposes a novel approach for efficiently solving an anisotropic transient diffusion problem using an enriched finite element method. We develop directional enrichment for the finite elements in the spatial discretization and a fully implicit scheme for the temporal discretization of the governing equations. Within this comprehensive framework, the proposed class of exponential functions as enrichment enhance the approximation of the finite element method by capturing the directional based behaviour of the solution. The incorporation of these enrichment functions leverages a priori knowledge about the anisotropic problem using the partition of unity technique, resulting in significantly improved approximation efficiency while retaining all the advantages of the standard finite element method. Consequently, the proposed approach yields accurate numerical solutions even on coarse meshes and with significantly fewer degrees of freedom compared to the standard finite element methods. Moreover, the choice of mesh coarseness remains independent of the anisotropy in the problem, enabling the use of the same mesh regardless of changes in the anisotropy. Using extensive numerical experiments, we consistently demonstrate the efficiency of the proposed method in attaining the desired levels of accuracy. Our approach not only provides reliable and precise solutions but also extends the capabilities of the finite element method to effectively address aspects of the heterogeneous anisotropic transient diffusion problems that were previously considered ineffective when using this method.

α-Robust Error Analysis of L2-1σ Scheme on Graded Mesh for Time-Fractional Nonlocal Diffusion Equation

Abstract In this work, a time-fractional nonlocal diffusion equation is considered. Based on the L2-1σ scheme on a graded mesh in time and the standard finite element method (FEM) in space, the fully-discrete L2-1σ finite element method is investigated for a time-fractional nonlocal diffusion problem. We prove the existence and uniqueness of fully-discrete solution. The α-robust error bounds are derived, i.e., bounds remain valid as α→1−, where α ∈(0,1) is the order of a temporal fractional derivative. The numerical experiments are presented to justify the theoretical findings.

The effect of peristalsis on dispersion in Casson fluid flow

The present study examines the peristaltic flow of a non-Newtonian (Casson) fluid and solute transport through a flexible tube. Using the long-wavelength approximation, an analytical solution for the Casson fluid velocity is obtained in the axial and radial directions. Then, a numerical solution to the solute concentration is computed using the standard Galerkin finite element method (FEM) for the case of solutal convection-diffusion and the Discontinuous Galerkin FEM for the pure convection case. Graphical results are presented for the axial and local solute concentrations, and the influence of peristaltic wave amplitude, volumetric flow rate, modified Péclet number, time, yield stress, and pertinent initial condition parameters were investigated.

Robust convergence result of discontinuous Galerkin stabilization method for two-dimensional reaction–diffusion equation with discontinuous source term

A reaction–diffusion problem with discontinuous source term and Dirichlet's boundary conditions on the unit square is considered in this paper. The proposed problem has been discretized using a combination of standard Galerkin finite element method (FEM) and non-symmetric discontinuous Galerkin finite element method with an interior penalty (NIPG) with bilinear elements. Layer adapted mesh of Shishkin type has been utilized to discretize the domain. Standard Galerkin FEM is applied on the layer part of the domain where the domain is dense enough and NIPG is applied to the outside layer part. By means of special choice of discontinuity-penalization parameters, the scheme is proved to be uniformly convergent of order O ( ε 1 / 4 N − 1 + N − 1 ln ⁡ N ) . Numerical tests are carried out in support of theoretical findings.

Free and forced vibration analysis over meshes with tangled (non-convex) elements

Tangled (non-convex) elements, i.e. elements with negative Jacobian determinant, can lead to erroneous results in the standard finite element method (FEM). Constructing tangle-free, well-structured meshes for complex geometries is often impossible. Hence there is a need to explore analysis methods that can directly handle such tangled meshes.In this paper, we propose the isoparametric tangled finite element method (i-TFEM) for free and forced vibration problems over tangled meshes. By employing piece-wise invertible mapping, a variational formulation is derived, leading to a simple modification of the standard FEM stiffness and mass matrices with the incorporation of additional compatibility constraints. Moreover, i-TFEM reduces to standard FEM for non-tangled (regular) meshes. The proposed method is implemented for three types of elements: 4-node quadrilateral, 9-node quadrilateral, and 8-node hexahedral elements. The numerical results demonstrate that i-TFEM is able to consistently handle general tangled (non-convex) elements, enabling convenient meshing for complex geometries.

NON-ISOTHERMAL MODELING OF SOIL VAPOR EXTRACTION SYSTEM INCLUDING SOIL TEMPERATURE EFFECT

Soil vapor extraction (SVE) is a proven effective in-situ technology for the removal of volatileorganic compounds (VOCs) from the subsurface. SVE process is highly sensitive to temperature.Studying annual soil temperature variation with depth declares that there is a considerable temperaturevariation in the upper few meters that may affect the overall efficiency of SVE process.A numerical model was developed to aid in investigation of field-scale soil vapor extractionprocess. The model is three-dimensional, time dependent that simulates nonisothermal vapor flow andtransport of multicomponent mixtures in soil and keeps track of the distribution of each compound inthe other three immobile phases (NAPL, aqueous, and sorbed). Rate limited interphase mass transferwith linear driving force expressions were used to model volatilization of oil into gas phase. A localequilibrium partitioning was assumed between gas, water, and solid phase. The model equations werediscretized using a standard Galerkin finite element method and solved using set iterative solutionalgorithm.Simulation of hypothetical field-scale problems was done. The physical domain described a threedimensionalsystem with flow to a single extraction well. A hypothetical soil temperature variationwith depth was incorporated with the model. The result of these simulations showed that thistemperature variation has a considerable effect on system efficiency and may play a role in optimumsystem configuration.

3D numerical simulation of an anisotropic bead type thermistor and multiplicity of solutions

We perform some 3D numerical experiments for the approximation of the solutions to a bead type thermistor problem. We consider the case of a diagonal anisotropic diffusion matrix whose jth entry is of the form |∂u/∂xj|pj−2∂u/∂xj, u being the temperature inside the thermistor and the exponents pj, 1≤j≤3, lie in the interval (1,+∞). We first show some existence results for different notions of solutions, prove a maximum principle for each type of solution, and study certain symmetry properties for these solutions in a bead type thermistor. These properties lead us to the introduction of a symmetric solution and we show the existence of such a solution.We have developed a numerical algorithm for the computation of the numerical solutions in a bead type thermistor. This algorithm combines a fixed-point technique with a standard finite element method (FEM). Some numerical tests have shown the existence of non-symmetric solutions and this leads to multiple many solutions (at least three). We discuss the numerical results obtained for different values of the exponents pj and the applied voltage on different meshes.

A scalable well-balanced numerical scheme for the simulation of fast landslides with efficient time stepping

We consider a single-phase depth-averaged model for the numerical simulation of fast-moving landslides with the goal of constructing a well-balanced, yet scalable and efficient, second-order time-stepping algorithm. We apply a Strang splitting approach to distinguish between parabolic and hyperbolic problems. For the parabolic contribution, we adopt a second-order Implicit-Explicit Runge-Kutta-Chebyshev scheme, while we use a two-stage Taylor discretization combined with a path-conservative strategy, to deal with the purely hyperbolic contribution. The proposed strategy allows to decouple hyperbolic from parabolic-reaction stiff contributions resulting in an overall well-balanced scheme subject just to stability restrictions of the hyperbolic term. The spatial discretization we adopt is based on a standard finite element method, associated with a hierarchically refined Cartesian grid. After providing numerical evidence of the well-balancing property, we demonstrate the capability of the proposed approach to select time steps larger than the ones adopted by a classical Taylor-Galerkin scheme. Finally, we provide some meaningful scaling results on ideal and realistic scenarios.

Numerical Approximation of Gaussian Random Fields on Closed Surfaces

Abstract We consider the numerical approximation of Gaussian random fields on closed surfaces defined as the solution to a fractional stochastic partial differential equation (SPDE) with additive white noise. The SPDE involves two parameters controlling the smoothness and the correlation length of the Gaussian random field. The proposed numerical method relies on the Balakrishnan integral representation of the solution and does not require the approximation of eigenpairs. Rather, it consists of a sinc quadrature coupled with a standard surface finite element method. We provide a complete error analysis of the method and illustrate its performances in several numerical experiments.

An implicit stabilized node-based smoothed finite element method for ultimate bearing capacity analysis of strip footing

The node-based smoothed finite element method (NS-FEM) has gained significant attention over the past decade. Nodal integration in NS-FEM, however, may lead to spurious oscillation of deformations of soil mass. To address the spurious oscillation of deformations associated with nodal integration, some stabilization techniques have been developed, but most of the stabilized NS-FEM have been implemented in the explicit finite element framework. The implicit framework, nonetheless, may offer some advantages including the high efficiency, good numerical accuracy, and suitability for handling large load increments and static problems. In this study, a novel stabilized NS-FEM by referring to the Taylor series expansion and incorporating the secant shear modulus, named sNS-FEM-KF(Gs), is developed in the implicit finite element framework. Based on the ultimate bearing capacity analysis of some strip footings, it is found that sNS-FEM-KF(Gs) may effectively eliminate the spurious oscillations observed in NS-FEM, and may resolve the overly-stiff behaviors of soil mass predicted by sNS-FEM-KF which is a reduced version of sNS-FEM-KF(Gs). Even compared to the standard finite element method with the 6-node triangular (T6) elements (i.e. FEM(T6)), sNS-FEM-KF(Gs) with linear 3-node triangular (T3) elements may outperform FEM(T6) in the investigated strip footing problems, indicating its good applicability in geotechnical analysis.

Convergence of finite element solution of stochastic Burgers equation

<abstract><p>We explore the numerical approximation of the stochastic Burgers equation driven by fractional Brownian motion with Hurst index $ H\in(1/4, 1/2) $ and $ H\in(1/2, 1) $, respectively. The spatial and temporal regularity properties for the solution are obtained. The given problem is discretized in time with the implicit Euler scheme and in space with the standard finite element method. We obtain the strong convergence of semidiscrete and fully discrete schemes, performing the error estimates on a subset $ \Omega_{k, h} $ of the sample space $ \Omega $ with the Gronwall argument being used to overcome the difficulties, caused by the subtle interplay of the nonlinear convection term. Numerical examples confirm our theoretical findings.</p></abstract>