Mesh deformation using the biharmonic operator

The use of the biharmonic operator for deforming a mesh in an arbitrary–Lagrangian–Eulerian simulation is investigated. The biharmonic operator has the advantage that two conditions can be specified on each boundary of the mesh. This allows both the position and the normal mesh spacing along a boundary to be controlled, which is important for two‐fluid interfaces and periodic boundaries. At these boundaries, we can simultaneously fix the position of the boundary and ensure that the normal mesh spacing is continuous across the boundary. In addition, results for deforming surfaces show that greater surface deformation can be tolerated when using biharmonic equations compared to approaches using second‐order partial differential equations. A final advantage is that with the biharmonic operator, the integrity of a grid in a moving boundary layer can be preserved as the boundary moves. The main disadvantage of the approach is its increased computational expense. Copyright © 2003 John Wiley & Sons, Ltd.     

Numerical solution of linear two-point boundary problems via the fundamental-matrix method

A numerical method is presented for the-solution of linear systems of differential equations with initial-value or two-point boundary conditions. For y ′(x) = A (x) y (x) + f (x) the domain of interest [a,b] is divided into an appropriate number L of subintervals. The coefficient matrix A (x) is replaced by its value A_k at a point x_k within the Kth subinterval, thus replacing the original system by the L discretized systems y k(x) = A _k y _k(x) + f _k(x), k = 1,2,…, L. The fundamental matrix solution Φ_k(x, x_k) over each subinterval is found by computing the eigenvalues and eigenvectors of each A _k. By matching the solutions y _k(x) at the L – 1 equispaced grid points defining the limits of the subintervals and the boundary conditions, the two-point problem is reduced to solving a system of linear algebraic equations for the matching constants characterizing the different y _k(x). The values of y ₁(a) and y _L(b) are used to calculate the missing boundary conditions. For initial-value problems this method is equivalent to a one-step method for generating approximate solutions. By means of a coordinate transformation, as in the multiple shooting method,¹ the method becomes particularly suitable for stiff systems of linear ordinary differential equations. Five examples are discussed to illustrate the viability of the method.     

An adaptive remeshing procedure for discontinuous finite element limit analysis

An adaptive remeshing procedure is proposed for discontinuous finite element limit analysis. The procedure proceeds by iteratively adjusting the element sizes in the mesh to distribute local errors uniformly over the domain. To facilitate the redefinition of element sizes in the new mesh, the interelements discontinuous field of elemental bound gaps is converted into a continuous field, ie, the intensity of bound gap, using a patch‐based approximation technique. An analogous technique is subsequently used for the approximation of element sizes in the old mesh. With these information, an optimized distribution of element sizes in the new mesh is defined and then scaled to match the total number of elements specified for each iteration in the adaptive remeshing process. Finally, a new mesh is generated using the advancing front technique. This adaptive remeshing procedure is repeated several times until an optimal mesh is found. Additionally, for problems involving discontinuous boundary loads, a novel algorithm for the generation of fan‐type meshes around singular points is proposed explicitly and incorporated into the main adaptive remeshing procedure. To demonstrate the feasibility of our proposed method, some classical examples extracted from the existing literary works are studied in detail.     

Adaptive tetrahedral mesh generation by constrained Delaunay refinement

This paper presents a tetrahedral mesh generation method for numerically solving partial differential equations using finite element or finite volume methods in three‐dimensional space. The main issues are the mesh quality and mesh size, which directly affect the accuracy of the numerical solution and the computational cost. Two basic problems need to be resolved, namely boundary conformity and field points distribution. The proposed method utilizes a special three‐dimensional triangulation, so‐called constrained Delaunay tetrahedralization to conform the domain boundary and create field points simultaneously. Good quality tetrahedra and graded mesh size can be theoretically guaranteed for a large class of mesh domains. In addition, an isotropic size field associated with the numerical solution can be supplied; the field points will then be distributed according to it. Good mesh size conformity can be achieved for smooth sizing informations. The proposed method has been implemented. Various examples are provided to illustrate its theoretical aspects as well as practical performance. Copyright © 2008 John Wiley & Sons, Ltd.     

An algorithm for triangulating smooth three-dimensional domains immersed in universal meshes

We describe an algorithm to recover a boundary-fitting triangulation for a bounded C²-regular domain immersed in a nonconforming background mesh of tetrahedra. The algorithm consists in identifying a polyhedral domain ω_h bounded by facets in the background mesh and morphing ω_h into a boundary-fitting polyhedral approximation Ω_h of Ω. We discuss assumptions on the regularity of the domain, on element sizes and on specific angles in the background mesh that appear to render the algorithm robust. With the distinctive feature of involving just small perturbations of a few elements of the background mesh that are in the vicinity of the immersed boundary, the algorithm is designed to benefit numerical schemes for simulating free and moving boundary problems. In such problems, it is now possible to immerse an evolving geometry in the same background mesh, called a universal mesh, and recover conforming discretizations for it. In particular, the algorithm entirely avoids remeshing-type operations and its complexity scales approximately linearly with the number of elements in the vicinity of the immersed boundary. We include detailed examples examining its performance.     

Vibration analysis of variable stiffness laminated composite sandwich plates

Abstract

This paper presents the free vibration analysis of a variable stiffness laminated composite sandwich plates. The fiber orientation angle of the face sheets (Skin) is assumed to vary linearly with the x-axis. The problem formulation is based on the higher-order shear deformation plate theory HDST C⁰ coupled with p-version of finite element method. The elements of the stiffness and mass matrices are calculated analytically. The sandwich plate is presented with a uniform mesh of four p-elements and the convergence properties are achieved by increasing the degree p of the hierarchical shape functions. A calculation program is developed to determine the fundamental frequencies for different physical and mechanical parameters such as plate thickness, core to face sheets thickness ratio, orientation angle of curvilinear fibers and boundary conditions. The results obtained show a good agreement with the available solutions in the literature. New comparison study of vibration response of laminated sandwich plate between the straight and curvilinear fibers is presented.     

Stress-driven diffusion in a deforming and evolving elastic circular tube of single component solid with vacancies

The title problem is considered for an elastic circular tube of inner radius A and outer radius B. The tube is made of a single component solid with vacancies as its second component. The mole fraction of the massive species is denoted by x ¹, while that of the vacancies by x ⁰ = 1 – x ¹. The tube is completely surrounded by vacuum, serving as a reservoir of vacancies. One of the standard elasticity boundary conditions is applied at time t = 0, when the composition is uniform. The ensuing coupled deformation and diffusion leads to the evolving of A(t), B(t) and x ¹(R, t) as functions of time. Since the single component solid is not in contact with its vapor or liquid, the diffusion boundary condition is always tied to the elasticity problem through a surface condition that involves the normal configurational traction. Our chemical potential has an energy density term that serves as a source in the interior and the boundary conditions for the diffusion problem are such that the time rates of boundary accretion Ȧ(t) and Ḃ(t) must simultaneously satisfy two dissipative inequalities, one governed by the gradient of the internal chemical potential and the other by the normal configurational traction.     

Global existence and blow-up analysis for a nonlinear diffusion equation with inner absorption and boundary flux

This article deals with a nonlinear diffusion equation with inner absorption u _t ?=?(log^σ(1?+?u)u _x ) _x ???λ(1?+?u)log ^p (1?+?u), in ?₊?×?(0,?+∞), subject to a logarithmic boundary flux ? log^σ(1?+?u)u _x (0,?t)?=?(1?+?u)log ^q (1?+?u)(0,?t), t?∈?(0,?+∞). We establish the critical global existence curve and give the asymptotic behaviour close to the blow-up time.     

Moving kriging interpolation and element‐free Galerkin method

A new formulation of the element‐free Galerkin (EFG) method is presented in this paper. EFG has been extensively popularized in the literature in recent years due to its flexibility and high convergence rate in solving boundary value problems. However, accurate imposition of essential boundary conditions in the EFG method often presents difficulties because the Kronecker delta property, which is satisfied by finite element shape functions, does not necessarily hold for the EFG shape function. The proposed new formulation of EFG eliminates this shortcoming through the moving kriging (MK) interpolation. Two major properties of the MK interpolation: the Kronecker delta property (?_I( s _J)=δ_IJ) and the consistency property (∑_Iⁿ?_I( x )=1 and ∑_Iⁿ?_I( x )x_Ii=x_i) are proved. Some preliminary numerical results are given. Copyright © 2002 John Wiley & Sons, Ltd.     

A multiobjective mesh optimization framework for mesh quality improvement and mesh untangling

We propose a multiobjective mesh optimization framework for mesh quality improvement and mesh untangling. Our framework combines two or more competing objective functions into a single objective function to be solved using one of various multiobjective optimization methods. Methods within our framework are able to optimize various aspects of the mesh such as the element shape, element size, associated PDE interpolation error, and number of inverted elements, but the improvement is not limited to these categories. The strength of our multiobjective mesh optimization framework lies in its ability to be extended to simultaneously optimize any aspects of the mesh and to optimize meshes with different element types. We propose the exponential sum, objective product, and equal sum multiobjective mesh optimization methods within our framework; these methods do not require articulation of preferences. However, the solutions obtained satisfy a sufficient condition of weak Pareto optimality. Experimental results show that our multiobjective mesh optimization methods are able to simultaneously optimize two or more aspects of the mesh and also are able to improve mesh qualities while eliminating inverted elements. We successfully apply our methods to real‐world applications such as hydrocephalus treatment and shape optimization. Copyright © 2013 John Wiley & Sons, Ltd.     

Hyperelliptic curves with reduced automorphism group A 5

We study genus g hyperelliptic curves with reduced automorphism group A ₅ and give equations y ² = f(x) for such curves in both cases where f(x) is a decomposable polynomial in x ² or x ⁵. For any fixed genus the locus of such curves is a rational variety. We show that for every point in this locus the field of moduli is a field of definition. Moreover, there exists a rational model y ² = F(x) or y ² = x F(x) of the curve over its field of moduli where F(x) can be chosen to be decomposable in x ² or x ⁵. While similar equations have been given in (Bujalance et al. in Mm. Soc. Math. Fr. No. 86, 2001) over , this is the first time that these equations are given over the field of moduli of the curve.     

Maximum-Likelihood Estimation of the Parameters of a Four-Parameter Generalized Gamma Population from Complete and Censored Samples

Consider the four-parameter generalized Gamma population with location parameter c, scale parameter a, shape/power parameter b, and power parameter p (shape parameter d = bp) and probability density function f(x; c, a, b, p) = p(x — c) ^bp–1 exp {–[(x – c)/a] ^p }/a ^bp Γ(b), where a, b, p > 0 and x ≥ c ≥ 0. The likelihood equations for parameter estimation are obtained by equating to zero the first partial derivatives, with respect to each of the four parameters, of the natural logarithm of the likelihood function for a complete or censored sample. The asymptotic variances and covariances of the maximum-likelihood estimators are found by inverting the information matrix, whose components are the limits, as the sample size n → ∞, of the negatives of the expected values of the second partial derivatives of the likelihood function with respect to the parameters. The likelihood equations cannot be solved explicitly, but an iterative procedure for solving them on an electronic computer is described. The results of applying this procedure to samples from Gamma, Weibull, and half-normal populations are tabulated, as are the asymptotic variances and covariances of the maximum-likelihood estimators.     

Fast Parallel Algorithms for Matrix Reduction to Normal Forms

 We investigate fast parallel algorithms to compute normal forms of matrices and the corresponding transformations. Given a matrix B in ℳ _n,n (K), where K is an arbitrary commutative field, we establish that computing a similarity transformation P such that F=P ^-1 BP is in Frobenius normal form can be done in ?C ² _K . Using a reduction to this first problem, a similar fact is then proved for the Smith normal form S(x) of a polynomial matrix A(x) in ℳ _n,m (K[x]); to compute unimodular matrices U(x) and V(x) such that S(x)=U(x)A(x)V(x) can be done in ?C ² _K . We get that over concrete fields such as the rationals, these problems are in ?C ². Using our previous results we have thus established that the problems of computing transformations over a field extension for the Jordan normal form, and transformations over the input field for the Frobenius and the Smith normal form are all in ?C ² _K . As a corollary we establish a polynomial-time sequential algorithm to compute transformations for the Smith form over K[x]. Received: February 29, 1996; revised version: August 29, 1997     

Low-field Magnetoresistance,Specific Heat and Magnetocaloric Effect in Sr Substituted Pr<Subscript>0.7</Subscript>Ca<Subscript>0.3</Subscript>MnO<Subscript>3</Subscript>

We investigate the effect of ionic size variation on the electrical and thermodynamic properties in a series of Pr_0.7Ca_0.3−x Sr _x MnO₃ (PCSMO) samples. The increase in Sr content results in an increase of the unit cell volume, as a bigger Sr²⁺ ion replaces the smaller Ca²⁺ ions. Resistivity measurements show that the increase in the Sr content also results in the induction of a metal–insulator transition (T _MI), which increases with increasing Sr content. The activation energy (E _a), calculated from the resistivity data, decreases with increasing Sr content confirming the metallic character. The effect of the magnetic field on resistivity and specific heat has also been studied.     

A modified collocation method and a penalty formulation for enforcing the essential boundary conditions in the element free Galerkin method

The Element free Galerkin method, which is based on the Moving Least Squares approximation, requires only nodal data and no element connectivity, and therefore is more flexible than the conventional finite element method. Direct imposition of essential boundary conditions for the element free Galerkin (EFG) method is always difficult because the shape functions from the Moving Least Squares approximation do not have the delta function property. In the prior literature, a direct collocation of the fictitious nodal values & u circ; used as undetermined coefficients in the MLS approximation, u ^h (x) [u ^h (x)=Φ·& u circ;], was used to enforce the essential boundary conditions. A modified collocation method using the actual nodal values of the trial function u ^h (x) is presented here, to enforce the essential boundary conditions. This modified collocation method is more consistent with the variational basis of the EFG method. Alternatively, a penalty formulation for easily imposing the essential boundary conditions in the EFG method with the MLS approximation is also presented. The present penalty formulation yields a symmetric positive definite system stiffness matrix. Numerical examples show that the present penalty method does not exhibit any volumetric locking and retains high rates of convergence for both displacements and strain energy. The penalty method is easy to implement as compared to the Lagrange multiplier method, which increases the number of degrees of freedom and yields a non-positive definite system matrix.     

Exterior stable domain segmentation integral equation method for scattering problems

This paper is concerned with the development of an exterior domain segmentation method for the solution of two- or three-dimensional time-harmonic scattering problems in acoustic media. This method, based on a variational localized, symmetric, boundary integral equation formulation leads, upon discretization, to a sparse system of algebraic equations whose solution requires only O(N) multiplications, where N is the number of unknown nodal pressures on the scatterer surface. The new procedure is analogous to the one developed recently¹ except that in the present formulation we avoid completely the use of the hypersingular operator, thereby reducing the computational complexity. Numerical experiments for a rigid circular cylindrical scatterer subjected to a plane incident wave serve to assess its accuracy for normalized wave numbers, ka, ranging from 0 to 30, both directly on the scatterer and in the far field, and to confirm that, contrary to standard boundary integral equation formulations, the present procedure is valid for critical frequencies.     

A mesh adaptation procedure for periodic domains

In this paper, an original technique is developed in order to build adaptive meshes on periodic domains. The new approach has the important property that it is code‐reused. The procedure is used against three different algorithms, namely, MAdLib ( Int. J. Numer. Meth. Engng 2000; in press), mmg (Proc. 17th Int. Meshing Roundtable, 2008) and the couple Yams (Rapport Technique RT‐0252, 2001) /Ghs3d (Proc. 8th Int. Meshing Roundtable, 1999). None of the latter algorithms needs to be adapted before it is applied to periodic domains. Some examples of adaptation are presented based on analytical, isotropic and anisotropic mesh‐size fields. Periodicity in translation and rotation both are considered. Finally, the mesh adaptation strategy is used in order to reduce the computational cost of a prediction of strain heterogeneity throughout a periodic polycrystalline aggregate deforming by dislocation slip. Copyright © 2011 John Wiley & Sons, Ltd.     

Numerical determination of a class of generalized stress intensity factors

A modification of the collocation method for the numerical solution of Cauchy-type singular integral equations appearing in plane elasticity and, especially, crack problems is proposed. This modification, based on a variable transformation, applies to the case when the unknown function of the singular integral equation behaves like A(x ? c)^α + B(x ? c)^β, where α < 0, 0 < β ? α < 1, near an endpoint c of the integration interval. In plane elasticity such a point is either a crack tip or a corner point of the boundary of the elastic medium. Thus the method seems to be quite efficient for the numerical evaluation of generalized stress intensity factors near such points. A successful application of the method to the classical plane elasticity problem of an antiplane shear crack terminating at a bimaterial interface was also made.     

Exact finite element solutions to the Helmholtz equation

For one-, two- and three-dimensional co-ordinate systems finite element matrices for the wave or Helmholtz equation are used to produce a single difference equation holding at any point of a regular mesh. Under homogeneous Dirichlet or Neumann boundary conditions, these equations are solved exactly. The eigenfunctions are the discrete form of sine or cosine functions and the eigenvalues are shown to be in error by a term of + O(h²ⁿ) where n is the order of the polynomial approximation of the wave function. The solutions provide the means of testing computer programs against the exact solutions and allow comparison with other difference schemes.     

An algebraic method for smoothing surface triangulations on a local parametric space

This paper presents a new procedure to improve the quality of triangular meshes defined on surfaces. The improvement is obtained by an iterative process in which each node of the mesh is moved to a new position that minimizes a certain objective function. This objective function is derived from algebraic quality measures of the local mesh (the set of triangles connected to the adjustable or free node). If we allow the free node to move on the surface without imposing any restriction, only guided by the improvement of the quality, the optimization procedure can construct a high‐quality local mesh, but with this node in an unacceptable position. To avoid this problem the optimization is done in the parametric mesh, where the presence of barriers in the objective function maintains the free node inside the feasible region. In this way, the original problem on the surface is transformed into a two‐dimensional one on the parametric space. In our case, the parametric space is a plane, chosen in terms of the local mesh, in such a way that this mesh can be optimally projected performing a valid mesh, that is, without inverted elements. Several examples and applications presented in this work show how this technique is capable of improving the quality of triangular surface meshes. Copyright © 2005 John Wiley & Sons, Ltd.