Coupled field formulation of exactly integrated membrane finite elements insensitive to distortion

Abstract

An efficient 4-noded membrane element tolerant and insensitive to mesh distortion is developed using kinematically coupling the in-plane displacements which satisfy the equilibrium equations in Cartesian coordinates and results in material-dependent coupled field interpolation functions making the element free from volumetric and shear locking. The proposed formulation provides a preliminary concept for the satisfaction of natural boundary conditions which is used to satisfy the strong patch test. The proposed element, passed several benchmark problems from the literature, compared well with elasticity solutions, passed the weak patch test and the inf-sup test for both compressible and incompressible cases.     

Superconvergence analysis of nonconforming finite element method for time-fractional nonlinear parabolic equations on anisotropic meshes

In this paper, we prove a novel result of the consistency error estimate with order $O\left(h^2\right)$ for $E{Q}_1^{rot}$ element (see Lemma 2) on anisotropic meshes. Then, a linearized fully discrete Galerkin finite element method (FEM) is studied for the time-fractional nonlinear parabolic problems, and the superclose and superconvergent estimates of order $O(\tau +h^2)$ in broken $H^1$-norm on anisotropic meshes are derived by using the proved character of $E{Q}_1^{rot}$ element, which improve the results in the existing literature. Numerical results are provided to confirm the theoretical analysis.     

About the gauge conditions arising in Finite Element magnetostatic problems

In this paper, we deal with some magnetostatic models considered in vector potential formulations and solved by a Finite Element solver. In order to ensure the uniqueness of the solution, a gauge condition has to be imposed, and several possibilities occur. Moreover, the source term has to be correctly defined to ensure a physically admissible solution. We show the equivalence between some of these choices for several kinds of boundary conditions. Moreover, we highlight their characteristic behaviors on some numerical benchmarks to illustrate our theoretical results.     

MAC finite difference scheme for Stokes equations with damping on non-uniform grids

In this paper, we consider the numerical methods for stationary Stokes equations with damping. The mark and cell(MAC) method has been applied to discretize the problem on non-uniform grids. We establish the LBB condition and the stability for the MAC scheme. Then we obtain the second order super-convergence in L2 norm for both velocity and pressure on non-uniform grids. We also obtain the second order convergence for some terms of H1 norm of the velocity, and the other terms of H1 norm are second order convergence on uniform grids. Numerical experiments using the MAC scheme show agreement of the numerical results with theoretical analysis.     

Variation of density of diesel and biodiesel mixtures in three different temperature ranges

The use of biodiesel fuel is becoming increasingly imperative nowadays and it is necessary to know the change of density. In this work the variation of density of diesel-biodiesel blends across the scale of 0–100% by adding each time 2% biodiesel and then measuring the density of three different temperatures (5, 15, 25°C) covering and the usual scale of temperatures the use of mixtures of diesel-biodiesel. Through the extraction of equations can be known in advance the relationship of density of diesel-biodiesel blend, and temperature that is used.     

Local and parallel finite element methods for the mixed Navier–Stokes/Darcy model

In this paper, the mixed Navier–Stokes/Darcy problem which describes a fluid flow filtrating through porous media is considered. Based on two-grid discretizations, two local and parallel finite element algorithms for solving this mixed model are proposed. Numerical analysis and experiments are presented to show the efficiency and effectiveness of the local and parallel finite element algorithms.     

An optimal control approach to the optical flow problem

The optical flow problem is reduced to an optimal control problem governed by a linear parabolic equation having the unknown velocity field (the optical flow) as drift term. This model is derived from a new assumption, that is, the brightness intensity is conserved on a moving pattern driven by a Gaussian stochastic process. The optimality conditions are deduced by a passage to the limit technique in an approximating optimal control problem introduced for a regularization purpose. Finally, the controller uniqueness is addressed.     

Solution of the point reactor kinetics equations with temperature feedback by the ITS2 method

The ITS2 method is used to solve the point-reactor kinetics equations in the integral formulation with arbitrary number of delayed neutron groups and Newtonian temperature feedback. The method is based on low-order Taylor series expansions of neutron density and reactivity functions and uses variable time steps to control the numerical instabilities resulting from the stiff nature of the governing equations. Time steps are determined through an analytic criterion relating their magnitudes to the maximum admissible truncation error in the neutron-density expansion series. Temperature feedback is included in the reactivity as a function of the neutron density for different input types, including step change with adiabatic temperature feedback and compensated ramp functions. An iterative procedure is applied to determine the time steps while simultaneously updating the reactivity function. Numerical results show the ITS2 method is highly accurate for solving point reactor dynamics problems with temperature feedback.     

A new version of Kovarik’s approximate orthogonalization algorithm without matrix inversion

In 1970 Kovarik proposed approximate orthogonalization algorithms. One of them (algorithm B) has quadratic convergence but requires at each iteration the inversion of a matrix of similar dimension to the initial one. An attempt to overcome this difficulty was made by replacing the inverse with a finite Neumann series expansion involving the original matrix and its adjoint. Unfortunately, this new algorithm loses the quadratic convergence and requires a large number of terms in the Neumann series which results in a dramatic increase in the computational effort per iteration. In this paper we propose a much simpler algorithm which, by using only the first two terms in a different series expansion, gives us the desired result with linear convergence. Systematic numerical experiments for collocation and Toeplitz matrices are also described.     

A numerical study of variable coefficient elliptic Cauchy problem via projection method

In this paper, we investigate a numerical method for the solution of an inverse problem of recovering lacking data on some part of the boundary of a domain from the Cauchy data on other part for a variable coefficient elliptic Cauchy problem. In the process, the Cauchy problem is transformed into the problem of solving a compact linear operator equation. As a remedy to the ill-posedness of the problem, we use a projection method which allows regularization solely by discretization. The discretization level plays the role of regularization parameter in the case of projection method. The balancing principle is used for the choice of an appropriate discretization level. Several numerical examples show that the method produces a stable good approximate solution.