Local error estimation and adaptive remeshing scheme for least-squares mixed finite elements

Following the theme of our previous work on least-squares finite elements [10,28], we describe an adaptive remeshing scheme using local residuals as the error indicator. This choice of indicator is natural (and exact at the element level!) in the norm associated with the corresponding least-squares statement. The remeshing strategy applied here involves mesh enrichment by point insertion in a Delaunay scheme. Several refined grids and error plots are included for a representative model elliptic boundary-value problem.     

Adaptive tetrahedral remeshing for modified solid models

During the engineering design process, a designed engineering component is usually repeatedly modified and analyzed to reach final design objectives, and completely mesh regeneration for each design change is very time consuming. An efficient remeshing approach for modified solid models using existing prior existing meshes is proposed in this paper to resolve this issue. It is mainly achieved via three main steps: different face (between the input model and its modification) identification, local destruction zone generation, and local mesh regeneration. In this approach, by carefully selecting a destruction zone to be removed from the original mesh model, a final geometric adaptive mesh model of the modified model is generated, which is very important for downstream analysis accuracy control. Additionally, the complex boundary of the variation region that needs to be remeshed is set up by merging the boundaries of modified region and the destruction zone without using complex intersection operations, which ensure the approach’s robustness. Experimental results on practical engineering parts are also shown to demonstrate the method’s performance.     

hp Adaptive finite elements based on derivative recovery and superconvergence

In this paper, we present a new approach to hp-adaptive finite element methods. Our a posteriori error estimates and hp-refinement indicator are inspired by the work on gradient/derivative recovery of Bank and Xu (SIAM J Numer Anal 41:2294?C2312, 2003; SIAM J Numer Anal 41:2313?C2332, 2003). For element ?? of degree p, R(? ^p u _hp ), the (piece-wise linear) recovered function of ? ^p u is used to approximate ${|\varepsilon|_{1,\tau} = |\hat{u}_{p+1} - u_{p}|_{1,\tau}}$ , which serves as our local error indicator. Under sufficient conditions on the smoothness of u, it can be shown that ${\|{\partial^{p}(\hat{u}_{p+1} - u_{p})\|_{0,\Omega}}}$ is a superconvergent approximation of ${\|(I - R){\partial}^p u_{hp}\|_{0,\Omega}}$ . Based on this, we develop a heuristic hp-refinement indicator based on the ratio between the two quantities on each element. Also in this work, we introduce nodal basis functions for special elements where the polynomial degree along edges is allowed to be different from the overall element degree. Several numerical examples are provided to show the effectiveness of our approach.     

A technique for improving the accuracy of quadrangular mixed finite elements for Darcy’s flow on heterogeneous domains

The mixed hybrid finite element (MHFE) method is well suited for the resolution of the Darcy’s flow on anisotropic and heterogeneous domains. However, the accuracy of the method can be altered for highly distorted meshes with non-convex quadrangles. Moreover, the standard quadrangular MHFE method can lead to non-physical oscillations for transient simulations since it does not respect the maximum principle.In this work, we derive a new realization of the MHFE method on general convex or non-convex quadrangular elements. Each element is fictitiously divided into triangles. The mass balance and flux law are then discretized over each triangle and aggregated to eliminate interior degrees of freedom at the quadrangular element level. The method is combined with the mass lumping procedure for triangles to improve the monotonicity of the discretization. The material properties as well as the pressure and the divergence of the flux are allowed to vary inside the quadrangular element to better describe heterogeneous domains.The obtained matrix is symmetric and positive definite and has the same stencil than the standard approach. The numerical experiments show the performances of this formulation compared to the standard one for heterogeneous domains and non-convex elements. An example is also provided for transient flow simulations where the unphysical oscillations are avoided with the new approach.     

Efficient tetrahedral remeshing of feature models for finite element analysis

Finite element analysis is nowadays widely used for product testing. At various moments during the design phase, aspects of the physical behaviour of the product are simulated by performing an analysis of the model. For each analysis, a mesh needs to be created that represents the geometry of the model at that point. In particular during the later stages of the development cycle, often only minor modifications are made to a model between design iterations. In that case it can be beneficial to reuse part of the previous mesh, especially if it was costly to construct. A new method is presented that efficiently constructs a tetrahedral mesh based on a tetrahedral mesh of a feature model at an earlier point of the design cycle. This is done by analysing the difference of the two feature models from the point of view of the individual features. By this means we can find a natural correspondence between the geometries of the feature models, and relate this to the mesh of the earlier model. We discuss the algorithm, gained improvements, quality of the results, and conditions for this method to be effective.     

A triangular finite element with local remeshing for the large strain analysis of axisymmetric solids

In this work, a three-node triangular finite element with two degrees of freedom per node for the large strain elasto-plastic analysis of axisymmetric solids is presented. The formulation resorts to the adjacent elements to obtain a quadratic interpolation of the geometry over a patch of four elements from which an average deformation gradient is defined. Thus, the element formulation falls within the framework of assumed strain elements or more precisely of F-bar type formulations. The in-plane behavior of the element is similar to the linear strain triangle, but without the drawbacks of the quadratic triangle, e.g. contact or distortion sensitivity. The element does not suffer of volumetric locking in problems with isochoric plastic flow and the implementation is simple. It has been implemented in a finite element code with explicit time integration of the momentum equations and tools that allow the simulation of industrial processes. The widely accepted multiplicative decomposition of the deformation gradient in elastic and plastic components is adopted here. An isotropic material with non-linear isotropic hardening has been considered. Two versions of the element have been implemented based on a Total and an Updated Lagrangian Formulation, respectively. Some approximations have been considered in the latter formulation aimed to reduce the number of operations in order to increase numerical efficiency. To consider bulk forming, with large geometric changes, an automatic local remeshing strategy has been developed. Several examples are considered to assess the element performance with and without remeshing.     

Estimating singularity powers with finite elements

Six-noded isoparametric finite elements are used to estimate singularity powers for problems with known solutions. The effect of crack length to crack tip element ratios as well as mesh refinement are shown to produce slight deviations in estimates of the singularity power. The bi-material problem with a crack terminating at the interface is investigated. Estimates of the singularity power for various material combinations are presented. A method of possibly improving singularity power estimations using finite elements is discussed.     

Adaptive finite elements for a certain class of variational inequalities of second kind

In this note, we extend our studies on finite element Galerkin schemes for elliptic variational inequalities of first to the one of second kind. Especially we perform the corresponding a posteriori error analysis for a simple friction problem and a model flow of a Bingham fluid.     

Polygonal surface remeshing with Delaunay refinement

Polygonal meshes are used to model smooth surfaces in many applications. Often these meshes need to be remeshed for improving the quality, density or gradedness. We apply the Delaunay refinement paradigm to design a provable algorithm for isotropic remeshing of a polygonal mesh that approximates a smooth surface. The proofs provide new insights and our experimental results corroborate the theory.     

Surface remeshing with robust high-order reconstruction

Remeshing is an important problem in variety of applications, such as finite element methods and geometry processing. Surface remeshing poses some unique challenges, as it must deliver not only good mesh quality but also good geometric accuracy. For applications such as finite elements with high-order elements (quadratic or cubic elements), the geometry must be preserved to high-order (third-order or higher) accuracy, since low-order accuracy may undermine the convergence of numerical computations. The problem is particularly challenging if the CAD model is not available for the underlying geometry, and is even more so if the surface meshes contain some inverted elements. We describe remeshing strategies that can simultaneously produce high-quality triangular meshes, untangling mildly folded triangles and preserve the geometry to high-order of accuracy. Our approach extends our earlier works on high-order surface reconstruction and mesh optimization by enhancing its robustness with a geometric limiter for under-resolved geometries. We also integrate high-order surface reconstruction with surface mesh adaptation techniques, which alter the number of triangles and nodes. We demonstrate the utilization of our method to meshes for high-order finite elements, biomedical image-based surface meshes, and complex interface meshes in fluid simulations.     

Some remarks on progress with finite elements

Information is the most valuable but least valued tool that professionals have. The amount of data in science and technology grows so rapidly that broad-coverage compilations cannot be maintained but concentrate on the coverage of specialized topics. The volume of finite element literature in the form of books, conference proceedings and journal papers, as well as a number of developed finite element codes, has been growing at a prodigious rate. It is almost impossible to be up to date with all the relevant information. A bibliometric study is presented; the author takes the number of published papers on finite elements as a measure of the research activity in the field of finite element techniques and investigates some engineering fields/topics where these techniques have been/are used.     

Improved equilibrium finite elements

A decisive improvement in the finite element equilibrium model is presented. A technique for construction of high precision, yet simple, single-sheet elements is demonstrated. Operating upon displacements, the proposed elements are directly suitable for automated computer handling. Application is made to two-dimensional problems. Computed results exhibits good agreement with known solutions and verify the destined rate of convergence as well. Application to mixed models, being essentially perturbed equilibrium models [1],is straightforward.     

Adaptive control with finite time persistency of excitation

An adaptive control algorithm for discrete-time SISO systems is presented. The main feature is the fact that it requires an external persistently exciting signal for a finite interval of time. In this way tracking of a class of signals (such as set points) can be achieved within an arbitrarily small error.     

Adaptive hierarchical boundary elements

A residual definition of error indicators and estimators is used. A p-and a h-implementation of an adaptive hierarchical boundary element method is presented.

The direct version of boundary elements based on the collocation method is reviewed and the way in which the residual boundary function is obtained is presented. The hierarchical definition of the interpolation and its advantages are discussed. Numerical interpolation to compute error estimators and indicators is established and its non-dimensionality is defined.     

Stabilized low-order finite elements for frictional contact with the extended finite element method

Contact problem suffers from a numerical instability similar to that encountered in incompressible elasticity, in which the normal contact pressure exhibits spurious oscillation. This oscillation does not go away with mesh refinement, and in some cases it even gets worse as the mesh is refined. Using a Lagrange multipliers formulation we trace this problem to non-satisfaction of the LBB condition associated with equal-order interpolation of slip and normal component of traction. In this paper, we employ a stabilized finite element formulation based on the polynomial pressure projection (PPP) technique, which was used successfully for Stokes equation and for coupled solid-deformation–fluid-diffusion using low-order mixed finite elements. For the frictional contact problem the polynomial pressure projection approach is applied to the normal contact pressure in the framework of the extended finite element method. We use low-order linear triangular elements (tetrahedral elements for 3D) for both slip and normal pressure degrees of freedom, and show the efficacy of the stabilized formulation on a variety of plane strain, plane stress, and three-dimensional problems.     

Functional programming for finite elements

The problems of software reliability can, to some extent, be attributed to the nature of conventional imperative programming languages (FORTRAN, Pascal, etc.). Functional programming languages, on the other hand, adopt a radically different approach where the only control structure is the recursive application of a pre-defined function. This gives the advantage that programs are mathematical expressions and can be treated formally as such for the purpose of correctness proofs. The suitability of the functional approach is examined by writing a standard finite element program in SASL, which is a purely functional language. The clarity and conciseness of the solution is noted, and although certain problems of efficiency and optimal algorithm design require further research, it is clear that the overall top-down design imposed by a functional language is beneficial to the computational engineer.     

Adaptive mesh refinement for shells with modified Ahmad elements

This paper presents a simple scheme for the generation of a quadrilateral element mesh for shells with arbitrary three-dimensional geometry. The present mesh generation scheme incorporates a normal mesh generator for generating a mesh in the two-dimensional plane and a specific mapping technique which maps the two-dimensional mesh onto the three-dimensional curved surface. As the mapping is a one-to-one mapping between the mesh in the plane and that on the curved surface, the resulting surface discretization is compatible with the local mesh parameters in two dimensions. This scheme is further combined, both with a sophisticated error estimate determined by using the best guess values of bending moments and membrane and transverse shear forces obtained from a previous solution, and an effective mesh refinement strategy established at an element level in order to complete an adaptive analysis for shell structures. Numerical examples are shown to illustrate the principles and procedure of the present adaptive analysis.