Linear Triangular Finite Elements Research Articles

Auto Mesh generation algorithm for the convex domain with the triangular elements

Mesh creation is one of the primary tasks in implementing the Finite Element Method (FEM) to solve two- and three-dimensional boundary value problems. However, the whole procedure becomes tedious and problematic when higher-order finite elements are employed to construct mesh and prepare corresponding element data. In this study, we strive to develop a versatile algorithm for discretizing the two-dimensional domain using linear, quadratic, and cubic triangular finite elements. The algorithm is developed based on n vertices (actual or more in number) that constitute the boundary of the domain, along with a computed (n+1)-th point such as the centroid. The algorithm, using this input, generates an initial mesh consisting of n triangular elements. Then, in every subsequent step, the algorithm increases the number of triangles in the mesh fourfold compared to the previous step. The algorithm we present here can employ (a) straight-sided (linear, quadratic, and cubic) triangular elements to generate the mesh for domains with polygonal boundaries and (b) both straight-sided and curved (quadratic and cubic) triangular elements with two straight sides and one curved side to generate meshes for domains with curved boundaries. Thus, the algorithm generates the desired meshes if the vertices are specified once at the initial step. Additionally, the inclusion of the mathematical expression of the curved boundary enables the algorithm to generate the fine mesh for the curved domain utilizing higher-order curved and straight-sided triangular elements. We also present a computer code in MATLAB incorporating the algorithm to create the mesh, prepare the element's data, and determine the element's connectivity.
 GANIT J. Bangladesh Math. Soc. 43.1 (2023) 017- 035

Solution of Grad-Shafranov equation with linear triangular finite elements providing magnetic field continuity with a second order accuracy

The formulation presented here is applied to the Grad-Shafranov equation, which describes the equilibrium of two-dimensional axisymmetric fusion plasmas. The equilibrium configurations, obtained via standard finite element formulations in terms of poloidal flux using linear triangles, provide the magnetic flux with an accuracy O(h2). However, the poloidal magnetic field is obtained with a lower accuracy O(h) and is discontinuous across adjacent elements, with consequent problems in the derivation of linearized models for Magneto-Hydro-Dynamic stability analysis. This requires the calculation of terms related to magnetic field derivatives. The method presented here, based on Helmholtz's theorem, achieves continuity and convergence rate of order O(h2) for the magnetic field by using three-node linear triangular finite elements. It can be implemented as a postprocessor of the magnetic flux solution obtained with linear triangles, in such a way to provide the magnetic field with an accuracy O(h2). The continuity properties of the magnetic field provide a high level of accuracy and the possibility of deriving reliable linearized models with a limited computational effort. The accuracy is highly reliable for some applications requiring a high precision in the vicinity of magnetic poloidal field null points, like for instance breakdown analysis, or control of the X-point of diverted configurations in a tokamak. The effectiveness of the method is shown in linear and nonlinear cases for which analytical solutions are available.

Novel superconvergence analysis of anisotropic triangular FEM for a multi‐term time‐fractional mixed sub‐diffusion and diffusion‐wave equation with variable coefficients

AbstractA fully discrete scheme is proposed for multi‐term time‐fractional mixed sub‐diffusion and diffusion‐wave equation with variable coefficients on anisotropic meshes, where linear triangular finite elelment method (FEM) is used for the spatial discretization and modified L1 approximation coupled with Crank–Nicolson scheme is applied to temporal direction. The mixed equation concerned contains a time–space coupled derivative which is very different from the previous literature. The stability is firstly obtained. Based on the property of the projection operator, the special relation between the projection operator and the interpolation operator of linear triangular finite element, the optimal error estimation and the superclose result are deduced. Then the global superconvergence property is derived by the interpolated postprocessing technique. Some numerical experiments are carried out to confirm the theoretical analysis on anisotropic meshes.

Study of Steady State Heat Conduction in Composite Layer by Linear Triangular Finite Element

Finite element method with linear triangular shape function was used to numerically investigate steady state heat conduction in composite layer. This method divides each layer into a number of small elements. By defining elements, local node, global node, and temperature boundary conditions, an element coefficient matrix can be formed. Element coefficient matrix needs to be assembled to form a global coefficient matrix. The solution of the global coefficient matrix will provide the temperature value at each node. In this research, variations of the thermal conductivity value and temperature boundary conditions are applied. The results of the study will show the effect of different thermal conductivity value against temperature gradient.

FEM-IDS for a second order strongly damped wave equation with memory

A linear triangular finite element method-implicit difference scheme (FEM-IDS) for the solution of second order strongly damped wave equation (SDWE) with memory on domain with interfaces is proposed. Sufficient conditions that guarantee the existence of a unique solution are given. The finite element discretization is such that the arbitrary (but smooth) interface is first approximated by a polygon with a boundary whose vertices all lie on the interface. With the interface being at σ-distance from the approximate interface, linear interpolation operator incorporating the influence of σ isproved. This together with Ritz–Volterra operator and some auxiliary error estimates in the neighborhood of the interface are used to obtain the convergence estimate of the proposed scheme. The scheme is applied to some test examples and the numerical results confirm the computational efficiency of the method.

A lower-bound formulation for unsaturated soils

A lower-bound formulation is proposed for consistent stability analysis of saturated and unsaturated soils with the suction stress-based effective stress employed to represent the shearing strength variation caused by matric suction. All constraints are expressed in terms of the effective stress. The body and surface forces induced by the suction stress are established and their effect on the equilibrium of unsaturated soils is studied in detail. Numerical solutions of the formulation are obtained by utilising linear triangular finite elements in combination with the second-order cone programming. Some numerical tests are given, and the lower-bound solutions are compared with the experimental data, the field measurements and the results of other methods. Close agreement is seen in all cases and the bounding property of the upper- and lower-bound formulations is guaranteed. The impact of the suction stress and the hydraulic state on the stability of some unsaturated earth structures is discussed.

Software for deformable solid mechanics

Subject of Research. The paper presents a software building method for the tasks of deformable solid mechanics. This software should guarantee high accuracy and speed of calculations, as well as simple preparation of the initial data and data processing even for an inexperienced user. The software was developed using the open source GMSH mesh generator application programming interface (API) and the Eigen mathematical library. Method. The developed software consists of three modules: GMSH_API, InputFile, FEMSolver and a database. The GMSH_API module, which prepares the finite element model, was written using the GMSH mesh generator API. The InputFile module describes methods for interacting with a previously created database, that provides quick and easy preparation of the input file needed to start the calculation. Numerical calculation by the finite element method is implemented in the FEMSolver module. The Eigen mathematical library was actively used for its implementation, and it can build sparse matrices that do not store zero elements in memory. This possibility obviates the need for additional transformations of the global stiffness matrix used in the finite element method. Main Results. The Kirsch task was solved as an example in a plane-stressed setting: a distributed tensile load was applied to the upper edge of a steel plate, with a round hole in the center, the lower edge of the plate was rigidly fixed. After calculating and obtaining the von Mises stress distribution field in the plate, we observe an error of 1.72% relative to the analytical solution. Such error value is considered low, therefore, the developed software not only facilitates the preparation of data for calculation, but also guarantees high accuracy of the obtained results. Practical Relevance. Commercial software for solving the problems of deformable solid mechanics, such as ANSYS Mechanical APDL, Abaqus, etc., is very expensive. Free software is primarily focused on researchers and, as a rule, is difficult for learning by an ordinary user-engineer, and the compromise version of the PDE Toolbox for MATHLAB is applicable only for tasks in a two-dimensional area and only supports a linear triangular finite element. However, the application of GMSH API and the Eigen library provides for creation of an easy-to-use but powerful tool for solving the problems of deformable solid mechanics.

High Spatial Accuracy Analysis of Linear Triangular Finite Element for Distributed Order Diffusion Equations

In this paper, an effective numerical fully discrete finite element scheme for the distributed order time fractional diffusion equations is developed. By use of the composite trapezoid formula and the well-known $L1$ formula approximation to the distributed order derivative and linear triangular finite element approach for the spatial discretization, we construct a fully discrete finite element scheme. Based on the superclose estimate between the interpolation operator and the Ritz projection operator and the interpolation post-processing technique, the superclose approximation of the finite element numerical solution and the global superconvergence are proved rigorously, respectively. Finally, a numerical example is presented to support the theoretical results.

Superconvergence analysis of FEM for 2D multi-term time fractional diffusion-wave equations with variable coefficient

The goal of this paper is to discuss high accuracy analysis of a fully-discrete scheme for 2D multi-term time fractional wave equations with variable coefficient on anisotropic meshes by approximating in space by linear triangular finite element method and in time by Crank-Nicolson scheme. The stability is firstly proved unconditionally. In the analysis of superclose properties, how to deal with the item for variable coefficient is the main difficulty. In order to do this, a new projection operator is defined and the relationship between the proposed projection operator and interpolation operator about linear triangular finite element is deduced. Consequently, the global superconvergence result is obtained by use of interpolation postprocessing technique. The numerical examples show that the proposed numerical method is highly accurate and computationally efficient.

A new approach of superconvergence analysis for nonlinear BBM equation on anisotropic meshes

A new approach of linear triangular finite element method (FEM) is developed for nonlinear BBM equation on anisotropic meshes. Based on the superclose estimate between the interpolation and Ritz projection of this element together with post-processing technique, the superclose and superconvergence results are derived for the semi-discrete and Crank–Nicolson fully-discrete schemes. Finally, a numerical example is carried out to demonstrate the theoretical analysis.

Numerical study of compositional compressible degenerate two-phase flow in saturated–unsaturated heterogeneous porous media

We study the convergence of a combined finite volume–nonconforming finite element scheme on general meshes for a partially miscible two-phase flow model in anisotropic porous media. This model includes capillary effects and exchange between the phases. The diffusion term, which can be anisotropic and heterogeneous, is discretized by piecewise linear nonconforming triangular finite elements. The other terms are discretized by means of a cell-centered finite volume scheme on a dual mesh. The relative permeability of each phase is decentered according to the sign of the velocity at the dual interface. The convergence of the scheme is proved thanks to an estimate on the two pressures which allows to show estimates on the discrete time and compactness results in the case of degenerate relative permeabilities. A key point in the scheme is to use particular averaging formula for the dissolution function arising in the diffusion term. We show also a simulation of hydrogen production in nuclear waste management. Numerical results are obtained by in-house numerical code.

Hierarchical electrochemical modeling and simulation of bio-hybrid interfaces

In this article we propose and investigate a hierarchy of mathematical models based on partial differential equations (PDE) and ordinary differential equations (ODE) for the simulation of the biophysical phenomena occurring in the electrolyte fluid that connects a biological component (a single cell or a system of cells) and a solid-state device (a single silicon transistor or an array of transistors). The three members of the hierarchy, ordered by decreasing complexity, are: (i) a 3D Poisson–Nernst–Planck (PNP) PDE system for ion concentrations and electric potential; (ii) a 2D reduced PNP system for the same dependent variables as in (i); (iii) a 2D area-contact PDE system for electric potential coupled with a system of ODEs for ion concentrations. The backward Euler method is adopted for temporal semi-discretization and a fixed-point iteration based on Gummel’s map is used to decouple system equations. Spatial discretization is performed using piecewise linear triangular finite elements stabilized via edge-based exponential fitting. Extensively conducted simulation results are in excellent agreement with existing analytical solutions of the PNP problem in radial coordinates and experimental and simulated data using simplified lumped parameter models.

Asymptotic Expansions and Extrapolations ofH1-Galerkin Mixed Finite Element Method for Strongly Damped Wave Equation

AbstractIn this paper, a high-accuracyH1-Galerkin mixed finite element method (MFEM) for strongly damped wave equation is studied by linear triangular finite element. By constructing a suitable extrapolation scheme, the convergence rates can be improved from𝒪(h) to𝒪(h3) both for the original variableuinH1(Ω) norm and for the actual stress variablep= ∇utinH(div;Ω) norm, respectively. Finally, numerical results are presented to confirm the validity of the theoretical analysis and excellent performance of the proposed method.

Superconvergence analysis of anisotropic linear triangular finite element for nonlinear Schrödinger equation

The main aim of this paper is to apply the simplest anisotropic linear triangular finite element to solve the nonlinear Schrödinger equation (NLS). Firstly, the error estimate and superclose property with order O ( h 2 ) about the Ritz projection are given based on an anisotropic interpolation property and high accuracy analysis of this element. Secondly, through establishing the relationship between the Ritz projection and interpolation, the superclose property of the interpolation is received. Thirdly, the global superconvergence with order O ( h 2 ) is derived by use of the interpolation post-processing technique. Finally, a numerical example is provided to verify the theoretical results. It is noteworthy that the main results obtained for anisotropic meshes herein cannot be deduced by only employing the interpolation or Ritz projection.

A combined finite volume–nonconforming finite element scheme for compressible two phase flow in porous media

We propose and analyze a combined finite volume---nonconforming finite element scheme on general meshes to simulate the two compressible phase flow in porous media. The diffusion term, which can be anisotropic and heterogeneous, is discretized by piecewise linear nonconforming triangular finite elements. The other terms are discretized by means of a cell-centered finite volume scheme on a dual mesh, where the dual volumes are constructed around the sides of the original mesh. The relative permeability of each phase is decentred according the sign of the velocity at the dual interface. This technique also ensures the validity of the discrete maximum principle for the saturation under a non restrictive shape regularity of the space mesh and the positiveness of all transmissibilities. Next, a priori estimates on the pressures and a function of the saturation that denote capillary terms are established. These stabilities results lead to some compactness arguments based on the use of the Kolmogorov compactness theorem, and allow us to derive the convergence of a subsequence of the sequence of approximate solutions to a weak solution of the continuous equations, provided the mesh size tends to zero. The proof is given for the complete system when the density of the each phase depends on its own pressure.

The Crouzeix-Raviart type nonconforming finite element method for the nonstationary Navier-Stokes equations on anisotropic meshes

This paper is devoted to study the Crouzeix-Raviart (C-R) type nonconforming linear triangular finite element method (FEM) for the nonstationary Navier-Stokes equations on anisotropic meshes. By introducing auxiliary finite element spaces, the error estimates for the velocity in the L2-norm and energy norm, as well as for the pressure in the L2-norm are derived.

Quantification of Three-Dimensional Surface Deformation using Global Digital Image Correlation

A novel method is presented to experimentally quantify evolving surface profiles. The evolution of a surface profile is quantified in terms of in-plane and out-of-plane surface displacements, using a Finite Element based Global Digital Image Correlation procedure. The presented method is applied to a case study, i.e. deformation-induced surface roughening during metal sheet stretching. The surface roughness was captured in-situ using a confocal optical profiler. The Global Digital Image Correlation method with linear triangular finite elements is applied to track the three-dimensional material movement from the measured height profiles. The extracted displacement fields reveal the full-field kinematics accompanying the roughening mechanism. Local deviations from the (average) global displacements are the result of the formation, growth, and stretching of hills and valleys on the surface. The presented method enables a full-field quantitative study of the surface height evolution, i.e. in terms of tracked surface displacements rather than average height values such as Root-Mean-Square or height-height correlation techniques. However, the technique does require that an initial surface profile, i.e. contrast, is present and that the contrast change between two measurements is minimal.

An a posteriori error estimator for an unsteady advection–diffusion–reaction problem

In this work we introduce an a posteriori error estimator, of the residual type, for the unsteady advection–diffusion–reaction problem. For the discretization in time we use an implicit Euler scheme and a continuous, piecewise linear triangular finite elements for the space together with a stabilized scheme. We prove that the approximation error is bounded, by above and below, by the error estimator. Using that, an adaptive algorithm is proposed, analyzed and tested numerically to prove the efficiency of our approach.

A new cumulative wavefront reconstructor for the Shack–Hartmann sensor

Abstract. We present a new direct algorithm for the reconstruction of the optical wavefront from the Shack–Hartmann sensor measurements in Single Conjugate Adaptive Optics (SCAO) systems. The computational complexity of this method which is based on linear triangular finite elements is linear in the number of unknowns. Moreover, due to its simple structure, the cumulative reconstructor is pipelinable and parallelizable, which makes the effective computation even faster. Error estimates are given in dependence of the number of the subapertures and the data noise.

Averaging of gradient in the space of linear triangular and bilinear rectangular finite elements

Abstract An averaging method for the second-order approximation of the values of the gradient of an arbitrary smooth function u = u(x 1, x 2) at the vertices of a regular triangulation T h composed both of rectangles and triangles is presented. The method assumes that only the interpolant Πh[u] of u in the finite element space of the linear triangular and bilinear rectangular finite elements from T h is known. A complete analysis of this method is an extension of the complete analysis concerning the finite element spaces of linear triangular elements from [Dalík J., Averaging of directional derivatives in vertices of nonobtuse regular triangulations, Numer. Math., 2010, 116(4), 619–644]. The second-order approximation of the gradient is extended from the vertices to the whole domain and applied to the a posteriori error estimates of the finite element solutions of the planar elliptic boundary-value problems of second order. Numerical illustrations of the accuracy of the averaging method and of the quality of the a posteriori error estimates are also presented.