A family of symmetric mixed finite elements for linear elasticity on tetrahedral grids

A family of stable mixed finite elements for the linear elasticity on tetrahedral grids are constructed,where the stress is approximated by symmetric H(div)-Pk polynomial tensors and the displacement is approximated by C-1-Pk-1polynomial vectors,for all k 4.The main ingredients for the analysis are a new basis of the space of symmetric matrices,an intrinsic H(div)bubble function space on each element,and a new technique for establishing the discrete inf-sup condition.In particular,they enable us to prove that the divergence space of the H(div)bubble function space is identical to the orthogonal complement space of the rigid motion space with respect to the vector-valued Pk-1polynomial space on each tetrahedron.The optimal error estimate is proved,verified by numerical examples.     

A family of mixed finite elements for the biharmonic equations on triangular and tetrahedral grids

Science China Mathematics - This paper introduces a new family of mixed finite elements for solving a mixed formulation of the biharmonic equations in two and three dimensions. The symmetric stress...     

Automatic parallel generation of tetrahedral grids by using a domain decomposition approach

An algorithm for the automatic parallel generation of three-dimensional unstructured grids based on geometric domain decomposition is proposed. A software package based on this algorithm is described. Examples of generating meshes for some application problems on a multiprocessor computer are presented. It is shown that the parallel algorithm can significantly (by a factor of several tens) reduce the mesh generation time. Moreover, it can easily generate meshes with as many as 5 × 10⁷ elements, which can hardly be generated sequentially. Issues concerning the speedup and the improvement of the efficiency of the computations and of the quality of the resulting meshes are discussed.     

Numerical convergence on adaptive grids for control volume methods

An adaptive technique for control‐volume methods applied to second order elliptic equations in two dimensions is presented. The discretization method applies to initially Cartesian grids aligned with the principal directions of the conductivity tensor. The convergence behavior of this method is investigated numerically. For solutions with low Sobolev regularity, the found L² convergence order is two for the potential and one for the flow density. The system of linear equations is better conditioned for the adaptive grids than for uniform grids. The test runs indicate that a pure flux‐based refinement criterion is preferable.© 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2008     

Statistical quality control for ternary ordinal quality data

The paper considers various aspects of statistical quality control by means of sample data received on a ternary ordinal scale. A new method for evaluating quality level and dispersion, free of any latent numerical scale assumptions, is proposed. The emphasis is on working with large samples, which enable the statistical analysis, estimation and control by the use of approximate analytical expressions of these measures to be considerably simplified. Two complementary studies demonstrate the usage of the proposed approach. Copyright © 2010 John Wiley & Sons, Ltd.     

Average adjacencies for tetrahedral skeleton-regular partitions

For any conforming mesh, the application of a skeleton-regular partition over each element in the mesh, produces a conforming mesh such that all the topological elements of the same dimension are subdivided into the same number of child-elements. Every skeleton-regular partition has associated special constitutive (recurrence) equations. In this paper the average adjacencies associated with the skeleton-regular partitions in 3D are studied. In three-dimensions different values for the asymptotic number of average adjacencies are obtained depending on the considered partition, in contrast with the two-dimensional case [J. Comput. Appl. Math. 140 (2002) 673]. In addition, a priori formulae for the average asymptotic adjacency relations for any skeleton-regular partition in 3D are provided.     

Efficient embeddings of grids into grids

In this paper we explore one-to-one embeddings of two-dimensional grids into their ideal two-dimensional grids. The presented results are optimal or considerably close to the optimum. For embedding grids into grids of smaller aspect ratio, we prove a new lower bound on the dilation matching a known upper bound. The edge-congestion provided by our matrix-based construction differs from the here presented lower bound by at most one. For embedding grids into grids of larger aspect ratio, we establish five as an upper bound on the dilation and four as an upper bound on the edge-congestion, which are improvements over previous results.     

Optimal strategies for water quality control

In this paper, the use of optimal control theory to obtain optimal strategies for the control of aquatic models is illustrated. Several types of control variables are used including the rate of nutrient application and the rates of change of nutrient concentration in both the phytoplankton and zooplankton populations. Techniques are given to show how optimal control theory can be applied to several models with different states and control variables constraints. Explicit expressions and optimality conditions are given for singular controls whenever they exist. Some numerical techniques are suggested to couple the optimal control parts in the proper sequence.     

Quadratic convergence for cell-centered grids

This work discusses some of the convergence properties of approximations defined on standard cell-centered finite difference grids. It is shown that the order of convergence is quadratic in the grid spacing for both uniform and nonuniform grids. This order of convergence cannot be improved upon, even if uniform point-distributed grids are used. It is concluded that order of convergence arguments do not favor point-distributed grid construction over the more physically reasonable cell-centered construction. The techniques used are elementary and rely entirely on Taylor series expansions. Other applications of these techniques, such as to local grid refinement, are indicated.     

Nonobtuse tetrahedral partitions

We show how to generate refinements of tetrahedral partitions, where no obtuse angles appear. Such partitions play an important role in deriving discrete maximum principles and maximum norm error estimates for the finite element method. © 2000 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 16: 327–334, 2000     

Mesh Spacing Estimates and Efficiency Considerations for Moving Mesh Systems

Adaptive numerical methods for solving partial differential equations (PDEs) that control the movement of grid points are called moving mesh methods. In this paper, these methods are examined in the case where a separate PDE, that depends on a monitor function, controls the behavior of the mesh. This results in a system of PDEs: one controlling the mesh and another solving the physical problem that is of interest. For a class of monitor functions resembling the arc length monitor, a trade off between computational efficiency in solving the moving mesh system and the accuracy level of the solution to the physical PDE is demonstrated. This accuracy is measured in the density of mesh points in the desired portion of the domain where the function has steep gradient. The balance of computational efficiency versus accuracy is illustrated numerically with both the arc length monitor and a monitor that minimizes certain interpolation errors. Physical solutions with steep gradients in small portions of their domain are considered for both the analysis and the computations.     

Sparse grids for boundary integral equations

Summary. The potential of sparse grid discretizations for solving boundary integral equations is studied for the screen problem on a square in . Theoretical and numerical results on approximation rates, preconditioning, adaptivity and compression for piecewise constant and linear sparse grid spaces are obtained. Received March 17, 1998 / Revised version received September 10, 1998     

Overlapping grids for the diffusion equation

Yiqi Qiu ^{We examine the use of nonmatching, overlapping grids for theapproximate solution of time-dependent diffusion problems withNeumann boundary conditions. This problem arises as a modelof the so-called well test analysis of oil and gas reservoirs,which has geometry modelling requirements that make overlappinggrids particularly suitable. We describe the problem and theoverlapping grid approximation, and then carry out a stabilityand convergence analysis in one space dimension (1D). We showthat for suitable schemes, stability is relatively easy to establishin much more general situations. Convergence is less easy togeneralise, but we demonstrate that 2D approximations appearto have the same convergence behaviour as their 1D counterparts.}     

Area control in generating smooth and convex grids over general plane regions

A new method (adaptive smoothness functional) to produce convex and smooth grids over general plane regions has been introduced in a recent work by the authors [10]; this method belongs to the variational grid generation approach. Theoretical results showing that a convex grid over a region is obtained when this method is applied, were presented; the basic assumption was that at least one convex grid exists. A procedure to control large cells (bilateral smoothness functional), in addition to smoothness and convexity, was also presented. Experimental results, showing the effectiveness of these methods, were reported; however, no theoretical results were reported assuring that the area control can be always exerted. This article continues the same line of research, introducing a new version of the bilateral smoothness functional that improves the control of large areas. Unlike the former method, theoretical results show the effectiveness of the new bilateral smoothness functional to exert such control. Optimal grids obtained with the new functional are compared with those reached using the older version, demostrating the improvement.     

Purely tetrahedral quadruple systems

An oriented tetrahedron is a set of four vertices and four cyclic triples with the property that any ordered pair of vertices is contained in exactly one of the cyclic triples. A tetrahedral quadruple system of order n (briefly TQS(n)) is a pair (X,B), where X is an nelement set and B is a set of oriented tetrahedra such that every cyclic triple on X is contained in a unique member of B. A TQS(n) (X,B) is pure if there do not exist two oriented tetrahedra with the same vertex set. In this paper, we show that there is a pure TQS(n) if and only if n = 2,4 (mod 6), n > 4, or n = 1,5 (mod 12). One corollary is that there is a simple two-fold quadruple system of order n if and only if n = 2,4 (mod 6) and n > 4, or n = 1,5 (mod 12). Another corollary is that there is an overlarge set of pure Mendelsohn triple systems of order n for n=1,3 (mod 6), n > 3, or n =0,4 (mod 12).     

Lattices on tetrahedral partitions

In this paper, four-pencil lattices on tetrahedral partitions are studied. The explicit representation of a lattice, based upon barycentric coordinates, enables us to extend the lattice from a single tetrahedron to a tetrahedral partition. It is shown that the number of degrees of freedom is equal to the number of vertices of the tetrahedral partition. The proof is based on a lattice split approach.      

The problem of singularity for planar grids

We consider simple graphs and their adjacency matrices. In [2], Rara (1996) gives methods of reducing graphs which simplify the procedure of computing the determinant of their adjacency matrices. We continue this subject matter and give a general method of reducing graphs. By the use of this method we define a formula for computing the determinant of any planar grid and in particular settle the problem of their singularity.     

Geometrically adaptive grids for stiff Cauchy problems

A new method for automatic step size selection in the numerical integration of the Cauchy problem for ordinary differential equations is proposed. The method makes use of geometric characteristics (curvature and slope) of an integral curve. For grids generated by this method, a mesh refinement procedure is developed that makes it possible to apply the Richardson method and to obtain a posteriori asymptotically precise estimate for the error of the resulting solution (no such estimates are available for traditional step size selection algorithms). Accordingly, the proposed methods are more robust and accurate than previously known algorithms. They are especially efficient when applied to highly stiff problems, which is illustrated by numerical examples.     

Multi-neighboring grids schemes for solving PDE eigen-problems

Instead of most existing postprocessing schemes,a new preprocessing approach,called multineighboring grids(MNG),is proposed for solving PDE eigen-problems on an existing grid G(Δ).The linear or multi-linear element,based on box-splines,are taken as the frst stage Kh1Uh=λh1Mh1Uh.In this paper,the j-th stage neighboring-grid scheme is defned asKh jUh=λh j Mh jUh,where Kh j:=Mh j 1Kh1and Mh jUh is to be found as a better mass distribution over the j-th stage neighboring-gridG(Δ),and Kh jcan be seen as an expansion of Kh1on the j-th neighboring-grid with respect to the(j 1)-th mass distribution Mh j 1.It is shown that for an ODE model eigen-problem,the j-th stage scheme with 2j-th order B-spline basis can reach2j-th order accuracy and even(2j+2)-th order accuracy by perturbing the mass matrix.The argument can be extended to high dimensions with separable variable cases.For Laplace eigen-problems with some 2-D and 3-D structured uniform grids,some 2j-th order schemes are presented for j 3.