A finite volume method for Stokes problems on quadrilateral meshes

We present a finite volume method for Stokes problems using the isoparametric $Q_1$–$Q_0$ element pair on quadrilateral meshes. To offset the lack of the $inf$–$sup$ condition, a jump term of discrete pressure (stabilizing term) is added to the continuity approximation equation. Thus, we establish a stabilized finite volume scheme on quadrilateral meshes. Then, based on some superclose estimates, we derive the optimal error estimates in the $H^1$- and $L_2$-norms for velocity and in the $L_2$-norm for pressure, respectively. Numerical examples are provided to illustrate our theoretical analysis. We emphasize that our work is the first time to propose and analyze a finite volume method for Stoke problems using isoparametric elements on quadrilateral meshes.     

Refinement and coarsening of surface meshes

This paper presents an adaptation scheme for surface meshes. Both refinement and coarsening tools are based upon local retriangulation. They can maintain the geometric features of the given surface mesh and its quality as well. A mesh gradation tool to smooth out large size differences between neighboring (in space) mesh faces and a procedure to detect and resolve self-intersections in the mesh are also presented. Both are driven by an octree structure and make use of the presented refinement tool.     

Automatic mesh generation and modification techniques for mixed quadrilateral and hexahedral element meshes of non-manifold models

A typical geometric model usually consists of both solid sections and thin-walled sections. Through using a suitable dimensional reduction algorithm, the model can be reduced to a non-manifold model consisting of solid portions and two-dimensional portions which represent the mid-surfaces of the thin-walled sections. It is desirable to mesh the solid entities using three-dimensional elements and the surface entities using two-dimensional elements. This paper proposes a robust scheme to automatically generate such a mesh of mixed two-dimensional and three-dimensional elements. It also ensures that the mesh is conforming at the interface of the non-manifold geometries. Different classes of problems are identified and their corresponding solutions are presented.     

An algorithm for automatic 2D quadrilateral mesh generation with line constraints

Finite element method (FEM) is a fundamental numerical analysis technique widely used in engineering applications. Although state-of-the-art hardware has reduced the solving time, which accounts for a small portion of the overall FEM analysis time, the relative time needed to build mesh models has been increasing. In particular, mesh models that must model stiffeners, those features that are attached to the plate in a ship structure, are imposed with line constraints and other constraints such as holes. To automatically generate a 2D quadrilateral mesh with the line constraints, an extended algorithm to handle line constraints is proposed based on the constrained Delaunay triangulation and Q-Morph algorithm. The performance of the proposed algorithm is evaluated, and numerical results of our proposed algorithm are presented.     

A new automated scheme of quadrilateral mesh generation for randomly distributed line constraints

This paper presents a new automated method of quadrilateral meshes with random line constraints which have not been fully considered in previous models. The authors developed a new looping scheme and a direct quadrilateral forming algorithm based on advanced front techniques. This generator overcomes the limitations of previous studies such as line constraint, unmeshed hole and mesh refinement. A qualitative test reveals that our algorithm is reliable and suitable at the field needed for very accurate results. The developed direct method to handle line-typed features automatically makes the multiple discretizations without any user interaction and modification.     

Intelligent finite element mesh generation

A knowledge-based and automatic finite element mesh generator (INTELMESH) for two-dimensional linear elasticity problems is presented. Unlike other approaches, the proposed technique incorporates the information about the object geometry as well as the boundary and loading conditions to generate an a priori finite element mesh which is more refined around the critical regions of the problem domain. INTELMESH uses a blackboard architecture expert system and the new concept of substracting to locate the critical regions in the domain and to assign priority and mesh size to them. This involves the decomposition of the original structure into substructures (or primitives) for which an initial and approximate analysis can be performed by using analytical solutions and heuristics. It then uses the concept of wave propagation to generate graded nodes in the whole domain with proper density distribution. INTELMESH is fully automatic and allows the user to define the problem domain with minimum amount of input such as object geometry and boundary and loading conditions. Once nodes have been generated for the entire domain, they are automatically connected to form well-shaped triangular elements ensuring the Delaunay property. Several examples are presented and discussed. When incorporated into and compared with the traditional approach to the adaptive finite element analysis, it is expected that the proposed approach, which starts the process with near optimal initial meshes, will be more accurate and efficient.     

Anisotropic quadrilateral mesh generation: An indirect approach

In this paper a new indirect approach is presented for anisotropic quadrilateral mesh generation based on discrete surfaces. The ability to generate grids automatically had a pervasive influence on many application areas in particularly in the field of Computational Fluid Dynamics. In spite of considerable advances in automatic grid generation there is still potential for better performance and higher element quality. The aim is to generate meshes with less elements which fit some anisotropy criterion to satisfy numerical accuracy while reducing processing times remarkably. The generation of high quality volume meshes using an advancing front algorithm relies heavily on a well designed surface mesh. For this reason this paper presents a new technique for the generation of high quality surface meshes containing a significantly reduced number of elements. This is achieved by creating quadrilateral meshes that include anisotropic elements along a source of anisotropy.     

A hybrid ant colony optimization approach for finite element mesh decomposition

This paper examines the application of the ant colony optimization algorithm to the partitioning of unstructured adaptive meshes for parallel explicit time-stepping finite element analysis. The concept of the ant colony optimization technique for finding approximate solutions to combinatorial optimization problems is described.The application of ant colony optimization for partitioning finite element meshes based on triangular elements is described.A recursive greedy algorithm optimization method is also presented as a local optimization technique to improve the quality of the solutions given by the ant colony optimization algorithm. The partitioning is based on the recursive bisection approach.The mesh decomposition is carried out using normal and predictive modes for which the predictive mode uses a trained multilayered feed-forward neural network which estimates the number of triangular elements that will be generated after finite elements mesh generation is carried out.The performance of the proposed hybrid approach for the recursive bisection of finite element meshes is examined by decomposing two mesh examples.     

Adaptive multiresolution for finite volume solutions of gas dynamics

Multiresolution analysis is used to improve the CPU and memory performance of a finite volume scheme. Departing from Harten's original scheme we present a fully adaptive scheme in the sense that at a given time, the solution is represented in a compressed form by a set of significant wavelet coefficients. Numerical benchmarks for Euler's system of compressible gas dynamics are performed on triangular meshes.     

Genetic algorithms in mesh optimization for visualization and finite element models

This paper investigates the use of a genetic algorithm (GA) to perform the large-scale triangular mesh optimization process. This optimization process consists of a combination of mesh reduction and mesh smoothing that will not only improve the speed for the computation of a 3D graphical or finite element model, but also improve the quality of its mesh. The GA is developed and implemented to replace the original mesh with a re-triangulation process. The GA features optimized initial population, constrained crossover operator, constrained mutation operator and multi-objective fitness evaluation function. While retaining features is important to both visualization models and finite element models, this algorithm also optimizes the shape of the triangular elements, improves the smoothness of the mesh and performs mesh reduction based on the needs of the user.     

Tetrahedral finite element mesh generation from NURBS solid models

In this paper, the development and the implementation of a tetrahedral meshing algorithm for generation of finite element meshes from NURBS solid models is presented. The meshing algorithm is based on a Delaunay technique, and makes use of some spatial data structures. The algorithm is capable of generating both uniform and varying size four-node and ten-node tetrahedral meshes. The algorithm has been implemented in a building block approach as part of a software library. It has been used as a practical tool in engineering design processes. Several representative test cases illustrate the effectiveness of the automatic solid mesh generator.     

A finite element method for investigating general elastic multi-structures

A finite element method is proposed for investigating the general elastic multi-structure problem, where displacements on bodies, longitudinal displacements on plates, longitudinal displacements and rotational angles on rods are discretized using conforming linear elements, transverse displacements on plates and rods are discretized respectively using TRUNC elements and Hermite elements of third order, and the discrete generalized displacement fields in individual elastic members are coupled together by some feasible interface conditions. The unique solvability of the method is verified by the Lax–Milgram lemma after deriving generalized Korn’s inequalities in some nonconforming element spaces on elastic multi-structures. The quasi-optimal error estimate in the energy norm is also established. Some numerical results are presented at the end.     

A finite element approach for multiphase fluid flow in porous media

The main scope of this work is to carry out a mathematical framework and its corresponding finite element (FE) discretization for the partially saturated soil consolidation modelling in presence of an immiscible pollutant. A multiphase system with the interstitial voids in the grain matrix filled with water (liquid phase), water vapour and dry air (gas phase) and with pollutant substances, is assumed. The mathematical model addressed in this work was developed in the framework of mixture theory considering the pollutant saturation-suction coupling effects. The ensuing mathematical model involves equations of momentum balance, energy balance and mass balance of the whole multiphase system. Encouraging outcomes were achieved in several different examples.     

Using objects to handle complexity in finite element software

Engineering software is becoming ever more complex. Finite element programs have sophisticated graphical input and output facilities, and are increasingly required to be linked to other software such as CAD or databases. The paper shows how an object oriented approach to finite element programming can be used to handle this complexity. This requires an approach that is very different from that adopted in more traditional programming. A foundation finite element class system is developed. This represents the essential data structure of the main finite element classes. It is then shown how this system can be used in a graphical model of two dimensional structures. The finite element system imposed no constraints on the development of the graphical model, yet could still be used easily. An important feature is that the nodes and elements are distributed around the graphical model, rather than being held centrally. For instance nodes may belong to points or lines of the graphical model. This means that the data structure used in the program more closely matches the way that the user of the program is likely to think.     

Adaptive multigrid for finite element computations in plasticity

The solution of the system of equilibrium equations is the most time-consuming part in large-scale finite element computations of plasticity problems. The development of efficient solution methods are therefore of utmost importance to the field of computational plasticity. Traditionally, direct solvers have most frequently been used. However, recent developments of iterative solvers and preconditioners may impose a change. In particular, preconditioning by the multigrid technique is especially favorable in FE applications.The multigrid preconditioner uses a number of nested grid levels to improve the convergence of the iterative solver. Prolongation of fine-grid residual forces is done to coarser grids and computed corrections are interpolated to the fine grid such that the fine-grid solution successively is improved. By this technique, large 3D problems, invincible for solvers based on direct methods, can be solved in acceptable time at low memory requirements. By means of a posteriori error estimates the computational grid could successively be refined (adapted) until the solution fulfils a predefined accuracy level. In contrast to procedures where the preceding grids are erased, the previously generated grids are used in the multigrid algorithm to speed up the solution process.The paper presents results using the adaptive multigrid procedure to plasticity problems. In particular, different error indicators are tested.     

A Newton-like finite element scheme for compressible gas flows

Semi-implicit and Newton-like finite element methods are developed for the stationary compressible Euler equations. The Galerkin discretization of the inviscid fluxes is potentially oscillatory and unstable. To suppress numerical oscillations, the spatial discretization is performed by a high-resolution finite element scheme based on algebraic flux correction. A multidimensional limiter of TVD type is employed. An important goal is the efficient computation of stationary solutions in a wide range of Mach numbers, which is a challenging task due to oscillatory correction factors associated with TVD-type flux limiters. A semi-implicit scheme is derived by a time-lagged linearization of the nonlinear residual, and a Newton-like method is obtained in the limit of infinite CFL numbers. Special emphasis is laid on the numerical treatment of weakly imposed characteristic boundary conditions. Numerical evidence for unconditional stability is presented. It is shown that the proposed approach offers higher accuracy and better convergence behavior than algorithms in which the boundary conditions are implemented in a strong sense.     

Topological improvement procedures for quadrilateral finite element meshes

This paper presents a set of procedures for improving the topology of unstructured quadrilateral finite element meshes. These procedures are based on the topology of the finite element mesh, and all operations act only on local regions of the mesh. The goal is to optimize the topology such that the smoothing process can produce the best possible element quality. Topological improvement procedures are presented both for elements that are interior to the mesh and for elements connected to the boundary. Also presented is a discussion of efficiency and optimal ordering of the procedures. Several example meshes are included to show the effectiveness of the current approach in improving element qualities in a finite element mesh.     

A smoothed finite element method for shell analysis

A four-node quadrilateral shell element with smoothed membrane-bending based on Mindlin-Reissner theory is proposed. The element is a combination of a plate bending and membrane element. It is based on mixed interpolation where the bending and membrane stiffness matrices are calculated on the boundaries of the smoothing cells while the shear terms are approximated by independent interpolation functions in natural coordinates. The proposed element is robust, computationally inexpensive and free of locking. Since the integration is done on the element boundaries for the bending and membrane terms, the element is more accurate than the MITC4 element for distorted meshes. This will be demonstrated for several numerical examples.