Formulation of a new error indicator for adaptive meshing with Hermite boundary elements

Postprocessing techniques and h-adaptive finite element-eigenproblem analysis

Grid redistribution based on measurable error indicators for the direct boundary element method

General three-dimensional solutions are given for the Laplace equation and Helmholtz equation in the cylindrical co-ordinate system, for processes which remain harmonic with time. The expressions for solution which are presented, not only for the magnetic and electrical field variables, but also for the higher-order vector potential, permit an analytical solution. The interrelationships between the field variables, the vector potential, and the higher-order vector potential are discussed. Several examples of engineering applications of the analytical calculation method are given.

Numerical simulation of incompressible flows using adaptive unstructured meshes and the pseudo-compressibility hypothesis

A dual integral formulation for the Helmholtz equation problem at a corner is derived by means of the contour approach around the singularity. It is discovered that employing the contour approach the jump term comes half and half from the free terms in the L and M kernel integrations, respectively, which differs from the limiting process from an interior point to a boundary point where the jump term is descended from the L kernel only. Thus, the definition of the Hadamard principal value for hypersingular integration at the collocation point of a corner is extended to a generalized sense for both the tangent and normal derivative of double layer potentials as compared to the conventional definition. The free terms of the six kernel functions in the dual integral equations for the Helmholtz equation at a corner have been examined. The kernel functions of the Helmholtz equation are quite different from those of the Laplace equation while the free terms of the Helmholtz equation are the same as those of the Laplace equation. It is worth to point out that the Laplace equation is a special case of the Helmholtz equation when the wave number approaches zero.

Boundary element procedure for computation of internal directional derivatives in homogeneous Laplace's problems solved by the finite element method

A BEM-based topology optimization for acoustic problems considering tangential derivative of sound pressure

The Boundary Element Method (BEM) is not local and generates a full and nonsymmetric matrix, therefore the matrix solution time could easily grow beyond acceptable limits. Furthermore, in many cases a decoupling between geometry description and solution approximationwould be advantageous and desirable. This work aims at offering the BEM a new approximation capability, which could reduce the number of degrees of freedom (dofs) needed, and at providing the desired decoupling. The method proposed here takes advantage of the less restrictive requirements in BEM compared with FEM for the field approximation. The base of the new approximation becomes what has been called “Spectral Element” (SE): a super-entity gathering many elements, which are used only for geometry description and numerical integration. Spectral shape functions are generated using smooth positive definite compact supported Radial Basis Functions (RBF). The method was applied to linear elasticity in 3D and numerical tests were performed for different typical cases in order to highlight advantages, disadvantages and limits of the proposed methodology. 1 Approximation field in the BEM 1.1 Some general properties of the “finite element approximation” Both BEM and FEM, in order to reduce the infinite dimensional space of the solution to a finite one, use the so called finite transform. If x represents the vector of space coordinates and u the approximation field, a set of “generalised degrees of freedom” in the form of a vector α must be chosen so that: u(x) = A(x)α (1) Boundary Elements XXVII 165 © 2005 WIT Press WIT Transactions on Modelling and Simulation, Vol 39, www.witpress.com, ISSN 1743-355X (on-line) In equation (1) A(x) represents the coordinate function matrix, which contains functions fixed in space. The finite elements appear when the continuum under analysis is ideally cut into “small” pieces, the elements of the mesh. In order to ensure that the values of the field in few specific points (nodes) belonging to an element govern the whole behaviour of u inside it, equation (1) must be imposed simultaneously in each of the n nodes of an element:   u1(x) u2(x) .. un(x)   = U = Cα =   A(x1) A(x2) .. A(xn)  α (2) Now if C is square, i.e. if the number of nodes n is equal to the number of α dofs, and is not singular, the following equations hold: u(x) = A(x)α = A(x)C−1U = Φ(x)U where Φ = A(C)−1 (3) Generally the Φ matrix is called shape matrix. It interpolates the dof of the element, represented by the nodal values, in between the nodes. In the FEM four general properties are needed for the functions in Φ, called shape functions, each of which belongs to a different node: 1. Completeness condition: in order to be able to represent a constant field, (first term in a Taylor series development of any field) the sum of all functions must be one in every point of the element:

We present an adaptive refinement strategy for the h-version of the boundary element method with weakly singular operators on surfaces. The model problem deals with the exterior Stokes problem, and thus considers vector functions. Our error indicators are computed by local projections onto ID subspaces defined by mesh refinement. These indicators measure the error separately for the vector components and allow for component-independent adaption. Assuming a saturation condition, the indicators give rise to an efficient and reliable error estimator. Also we describe how to deal with meshes containing quadrilaterals which are not shape regular. The theoretical results are underlined by numerical experiments. To justify the saturation assumption, in the Appendix we prove optimal lower a priori error estimates for edge singularities on uniform and graded meshes.

In this paper, two-dimensional flows of viscous and viscoelastic fluids are simulated using radial-basis-function networks (RBFNs) and Cartesian grids. To solve the governing equations, high-order approximations based on one-dimensional integrated-RBFNs are employed to approximate the field variables and the Galerkin discretisation formulation is utilised to transform the differential equation into a set of algebraic equations. The main distinguishing feature of the above combination is that two processes, namely the representation of the field variables and the discretisation of the governing equations, are both based on integration. In sharp contrast to conventional Galerkin methods, derivative boundary conditions are presently imposed in an exact manner. Two test problems, namely natural convection of a Newtonian fluid in a concentric annulus and fully-developed flow of an Criminale-Ericksen-Filbey (CEF) fluid in a rectangular duct, are studied. Results obtained are compared with published data in the literature.

We present a finite volume adaptive mesh refinement method for solid-liquid phase change problems with convection. The refinement criterion consisted of three different error estimators for the solid-liquid interface, the flow field, and the temperature field respectively. For the solid-liquid interface, the cells undergoing phase change were refined based on the maximum difference in the liquid fraction over the cell faces. For the flow field and the temperature field, an error indicator was used based on the cell residual method. To maintain a high parallelization efficiency, a dynamic load balancing procedure was used. The adaptive mesh refinement strategy was verified through three different test cases, these are the gallium melting in both 2D and 3D cavities, and the molten salt reactor freeze valve. For all three cases, very good agreement was obtained between the adaptive mesh results and the reference solutions. In addition, more accurate results were obtained with the adaptive meshes compared to static meshes with a similar amount of mesh cells. This illustrated the potential of the current approach for developing computationally efficient numerical methods for solid-liquid phase change problems.

Some aspects of adaptive grid technology related to boundary and interior layers

Boundary element formulation of building slabs

A simplified two-dimensional boundary element method with arbitrary uniform mean flow

In the present work, we discuss the thermoelastic interactions in a half-space in the context of the theory of dual phase-lag thermoelasticity due to stochastic conditions applied at the boundary. We consider a one-dimensional problem with traction-free boundary subjected to two types of time-dependent thermal distributions. The boundary conditions are assumed to be random in nature to make the problem more realistic and the noise added is considered to be white noise. The problem is solved using Laplace transform and short time approximation is used to find the approximate solution for field variables while taking the inverse transforms. The solution for the stochastic case has been obtained by using the concept of a Wiener process and stochastic simulation. Numerical analysis based on stochastic simulation is carried out using copper material along different sample paths and comparative analysis between the deterministic and the stochastic distributions of field variables is presented. Special attention is paid to highlight the effects of considering randomness added to the boundary conditions.