A Collocation Code for Singular Boundary Value Problems in Ordinary Differential Equations

We present a MATLAB package for boundary value problems in ordinary differential equations. Our aim is the efficient numerical solution of systems of ODEs with a singularity of the first kind, but the solver can also be used for regular problems. The basic solution is computed using collocation methods and a new, efficient estimate of the global error is used for adaptive mesh selection. Here, we analyze some of the numerical aspects relevant for the implementation, describe measures to increase the efficiency of the code and compare its performance with the performance of established standard codes for boundary value problems.     

One- and two-level Schwarz methods for variational inequalities of the second kind and their application to frictional contact

We present and analyze subspace correction methods for the solution of variational inequalities of the second kind and apply these theoretical results to non smooth contact problems in linear elasticity with Tresca and non-local Coulomb friction. We introduce these methods in a reflexive Banach space, prove that they are globally convergent and give error estimates. In the context of finite element discretizations, where our methods turn out to be one- and two-level Schwarz methods, we specify their convergence rate and its dependence on the discretization parameters and conclude that our methods converge optimally. Transferring this results to frictional contact problems, we thus can overcome the mesh dependence of some fixed-point schemes which are commonly employed for contact problems with Coulomb friction.     

Method for constructing meshes adapting to the solution of boundary value problems for ordinary differential equations of the second and fourth orders

To solve boundary value problems for ordinary differential equations of the second and fourth orders, we suggest a method for constructing a sequence of adaptively refined and coarsened meshes in a version of the finite element method with piecewise cubic Hermite basis functions. The construction of the meshes is based on C-norm estimates of the variation in the approximate solution or in the value of the functional to be minimized under the addition of a test point to a mesh interval or the deletion of a point from the current mesh. We present the results of numerical experiments used to assess the efficiency of the method. By way of example, problems whose solutions have singularities of the boundary layer type were used in these experiments. We carry out a comparison with a version of the method based on the uniform mesh refinement.     

Anisotropic mesh adaptation for numerical solution of boundary value problems

We present an efficient mesh adaptation algorithm that can be successfully applied to numerical solutions of a wide range of 2D problems of physics and engineering described by partial differential equations. We are interested in the numerical solution of a general boundary value problem discretized on triangular grids. We formulate a necessary condition for properties of the triangulation on which the discretization error is below the prescribed tolerance and control this necessary condition by the interpolation error. For a sufficiently smooth function, we recall the strategy how to construct the mesh on which the interpolation error is below the prescribed tolerance. Solving the boundary value problem we apply this strategy to the smoothed approximate solution. The novelty of the method lies in the smoothing procedure that, followed by the anisotropic mesh adaptation (AMA) algorithm, leads to the significant improvement of numerical results. We apply AMA to the numerical solution of an elliptic equation where the exact solution is known and demonstrate practical aspects of the adaptation procedure: how to control the ratio between the longest and the shortest edge of the triangulation and how to control the transition of the coarsest part of the mesh to the finest one if the two length scales of all the triangles are clearly different. An example of the use of AMA for the physically relevant numerical simulation of a geometrically challenging industrial problem (inviscid transonic flow around NACA0012 profile) is presented. © 2004 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2004.     

Static contact analysis of spiral bevel gear based on modified VFIFE (vector form intrinsic finite element) method

Since the intrinsic limitations of FEM (Finite element method) and lumped-mass method, we derive the formula of 8-node hexahedral element based on VFIFE (vector form intrinsic finite element method) method and applied it in contact analysis of gears. This paper proposed a new method to determine pure nodal deformation, which could simplify the computation compared to the traditional VFIFE method. Combining the VFIFE method and matching contact algorithm, we analyzed spiral bevel gear meshing problems. Spiral bevel models with two different mesh densities are calculated analyzed by the VFIFE method and FEM. Performance indicators of gears are extracted and compared, including contact forces, contact and bending stresses, contact stress patterns and loaded transmission errors. The results show that the VFIFE method has a stable performance and reliable accuracy under coarse or refined mesh conditions, while the FEM inaccurately calculates the contact stress of the coarse mesh model. The examples demonstrate that the proposed method could precisely analyze gear meshing problems with a coarse mesh model, which provides a new solution for gear mechanics.     

求解含源项非定常对流扩散问题的移动网格方法

本文研究了一类含源项非定常奇异摄动对流扩散问题.利用Crank-Nicolson差分格式下的移动网格方法,获得了求解该类问题的数值实验结果,改进了均匀网格下求解的结果.     

High‐order numerical methods for one‐dimensional parabolic singularly perturbed problems with regular layers

In this work we construct and analyze some finite difference schemes used to solve a class of time‐dependent one‐dimensional convection‐diffusion problems, which present only regular layers in their solution. We use the implicit Euler or the Crank‐Nicolson method to discretize the time variable and a HODIE finite difference scheme, defined on a piecewise uniform Shishkin mesh, to discretize the spatial variable. In both cases we prove that the numerical method is uniformly convergent with respect to the diffusion parameter, having order near two in space and order one or 3/2, depending on the method used, in time. We show some numerical examples which illustrate the theoretical results, in the case of using the Euler implicit method, and give better numerical behaviour than that predicted theoretically, showing order two in time and order N^?2log²N in space, if the Crank‐Nicolson scheme is used to discretize the time variable. Finally, we construct a numerical algorithm by combining a third order A‐stable SDIRK with two stages and a third‐order HODIE difference scheme, showing its uniformly convergent behavior, reaching order three, up to a logarithmic factor. © 2004 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2005     

Enriched mixed numerical manifold formulation with continuous nodal gradients for dynamics of fractured poroelasticity

Based on the three-variable Biot model, a numerical manifold model is presented to investigate dynamic responses of the fractured poroelasticity, especially the interaction between hydro-mechanical wave and fracture. The most flexible 3-node triangular mesh serves as the mathematical cover (MC) of the present model regardless of the problem types, shunning difficulties in qualified mesh generation. Continuous nodal gradients and Kronecker-delta property are achieved for the global approximations by constructing the local approximations with a constrained and orthonormalized least square (CO-LS) scheme, presenting more precise effective stress fields and more convenience in implementing boundary conditions for both solid and fluid phases. Incorporating a stick-slip frictional contact model via the augmented Lagrange multiplier method, the present model is capable of accurately predicting the contact phenomenon in fractured poroelasticity. In terms of the energy balance condition, precision and stability of the proposed model in time integration are verified. Fractured porous media involved multiple cracks can be addressed more naturally and conveniently with the present model relative to extended finite element method (XFEM) and phantom node method (PNM). By solving a set of typical porous media problems, the superiority, accuracy and robustness of the present model are verified.     

Variational theory and domain decomposition for nonlocal problems

In this article we present the first results on domain decomposition methods for nonlocal operators. We present a nonlocal variational formulation for these operators and establish the well-posedness of associated boundary value problems, proving a nonlocal Poincaré inequality. To determine the conditioning of the discretized operator, we prove a spectral equivalence which leads to a mesh size independent upper bound for the condition number of the stiffness matrix. We then introduce a nonlocal two-domain variational formulation utilizing nonlocal transmission conditions, and prove equivalence with the single-domain formulation. A nonlocal Schur complement is introduced. We establish condition number bounds for the nonlocal stiffness and Schur complement matrices. Supporting numerical experiments demonstrating the conditioning of the nonlocal one- and two-domain problems are presented.     

A Comparative Numerical Study of Meshing Functionals for Variational Mesh Adaptation

We present a comparative numerical study for three functionals used for variational mesh adaptation. One of them is a generalization of Winslow's variable diffusion functional while the others are based on equidistribution and alignment. These functionals are known to have nice theoretical properties and work well for most mesh adaptation problems either as a stand-alone variational method or combined within the moving mesh framework. Their performance is investigated numerically in terms of equidistribution and alignment mesh quality measures. Numerical results in 2D and 3D are presented.     

Planar mesh refinement cannot be both local and regular

We show that two desirable properties for planar mesh refinement techniques are incompatible. Mesh refinement is a common technique for adaptive error control in generating unstructured planar triangular meshes for piecewise polynomial representations of data. Local refinements are modifications of the mesh that involve a fixed maximum amount of computation, independent of the number of triangles in the mesh. Regular meshes are meshes for which every interior vertex has degree 6. At least for some simple model meshing problems, optimal meshes are known to be regular, hence it would be desirable to have a refinement technique that, if applied to a regular mesh, produced a larger regular mesh. We call such a technique a regular refinement. In this paper, we prove that no refinement technique can be both local and regular. Our results also have implications for non-local refinement techniques such as Delaunay insertion or Rivara's refinement. Received August 1, 1996 / Revised version received February 28, 1997     

A finite element method for contact/impact

Ideas from the analysis of differential-algebraic equations are applied to the numerical solution of frictionless contact/impact problems in solid mechanics. Index-one and two formulations for dynamic contact–impact within the context of the finite element method are derived. The resulting equations are shown to stabilize the kinematic fields at the contact interface, at the expense of a small energy loss, which is shown to decrease consistently with mesh refinement. This energy dissipation is shown to be necessary for the establishment of persistent contact. A Newmark-type time integration scheme is derived from the proposed formulation, and shown to yield excellent results in modeling the transition to contact/impact.     

On the uniform convergence of a finite difference scheme for time dependent singularly perturbed reaction-diffusion problems

In this work we are interested in the numerical approximation of 1D parabolic singularly perturbed problems of reaction-diffusion type. To approximate the multiscale solution of this problem we use a numerical scheme combining the classical backward Euler method and central differencing. The scheme is defined on some special meshes which are the tensor product of a uniform mesh in time and a special mesh in space, condensing the mesh points in the boundary layer regions. In this paper three different meshes of Shishkin, Bahkvalov and Vulanovic type are used, proving the uniform convergence with respect to the diffusion parameter. The analysis of the uniform convergence is based on a new study of the asymptotic behavior of the solution of the semidiscrete problems, which are obtained after the time discretization by the Euler method. Some numerical results are showed corroborating in practice the theoretical results on the uniform convergence and the order of the method.     

Mixed Collocation for Fractional Differential Equations

We present the mixed collocation method for numerical integration of fractional differential equations of the type D ^β u=Φ(u,t). Given a regular mesh with constant discretization step, the unknown u(t) is considered as continuous and affine in each cell, and the dynamics Φ(u,t) as a constant. After a fractional integration, the equation is written strongly at the mesh vertices and the dynamics weakly in each cell. The “Semidif” software has been developed for the particular case of numerical integration of order 1/2. The validation for analytical results and published solutions is established and experimental convergence as the mesh size tends to zero is obtained. Good results are obtained for a nonlinear model with a strong singularity.     

Local and parallel finite element algorithm for stationary incompressible magnetohydrodynamics

This article presents a local and parallel finite element method for the stationary incompressible magnetohydrodynamics problem. The key idea of this algorithm comes from the two‐grid discretization technique. Specifically, we solve the nonlinear system on a global coarse mesh, and then solve a series of linear problems on several subdomains in parallel. Furthermore, local a priori estimates are obtained on a general shape regular grid. The efficiency of the algorithm is also illustrated by some numerical experiments.© 2017 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 1513–1539, 2017     

Unstructured mesh finite difference/finite element method for the 2D time-space Riesz fractional diffusion equation on irregular convex domains

Fractional differential equations are powerful tools to model the non-locality and spatial heterogeneity evident in many real-world problems. Although numerous numerical methods have been proposed, most of them are limited to regular domains and uniform meshes. For irregular convex domains, the treatment of the space fractional derivative becomes more challenging and the general methods are no longer feasible. In this work, we propose a novel numerical technique based on the Galerkin finite element method (FEM) with an unstructured mesh to deal with the space fractional derivative on arbitrarily shaped convex and non-convex domains, which is the most original and significant contribution of this paper. Moreover, we present a second order finite difference scheme for the temporal fractional derivative. In addition, the stability and convergence of the method are discussed and numerical examples on different irregular convex domains and non-convex domains illustrate the reliability of the method. We also extend the theory and develop a computational model for the case of a multiply-connected domain. Finally, to demonstrate the versatility and applicability of our method, we solve the coupled two-dimensional fractional Bloch–Torrey equation on a human brain-like domain and exhibit the effects of the time and space fractional indices on the behaviour of the transverse magnetization.     

Accurate modeling of contact using cubic splines

In this paper, we develop and implement a new method for the accurate representation of contact surfaces. This approach overcomes the difficulties arising from the use of traditional node-to-linear surface contact algorithms. In our proposed method, contact surfaces were modeled accurately using C¹-continuous cubic splines, which interpolate the finite element nodes. In this case, the unit normal vectors are defined uniquely at any point on the contact surfaces. These splines preserve the local deformation of the nodes on each flexible contact surface. Consequently, a consistent linearization of the kinematic contact constraints, based on the spline interpolation, was derived. Moreover, the gap between two contact surfaces was modeled accurately using an efficient surface-to-surface contact search algorithm. Since the continuity of the splines is not affected by the number of nodes, accurate stress distribution can be obtained with less finite elements at the contact surface than that using the traditional linear discretization of the contact surface. Two numerical examples are used to illustrate the advantages of the proposed representation. They show a significant improvement in accuracy compared to traditional piecewise element-based surface interpolation. This approach overcomes the problem of mismatch in a finite element mesh. This is very useful, since most realistic engineering problems involve contact areas that are not known a priori.     

On the solution of Maxwell's equations in polygonal domains

This paper is concerned with the structure of the singular and regular parts of the solution of time‐harmonic Maxwell's equations in polygonal plane domains and their effective numerical treatment. The asymptotic behaviour of the solution near corner points of the domain is studied by means of discrete Fourier transformation and it is proved that the solution of the boundary value problem does not belong locally to H² when the boundary of the domain has non‐acute angles. A splitting of the solution into a regular part belonging to the space H², and an explicitly described singular part is presented. For the numerical treatment of the boundary value problem, we propose a finite element discretization which combines local mesh grading and the singular field methods and derive a priori error estimates that show optimal convergence as known for the classical finite element method for problems with regular solutions. Copyright © 2006 John Wiley & Sons, Ltd.