An adaptive method for the determination of the order of element for acquiring the desired accuracy in 2D elastostatic FEM analysis

An adaptive method for the determination of the order of element (or element order) was developed for the finite element analysis of 2D elastostatic problems. Here, the order of element means the order of the polynomial function that interpolates the displacement distribution in the element. This method was based on acquiring the desired accuracy for each finite element. From the numerical experiments, the relationship ξ=k(1/p)^β was deduced, where ξ is the error of the result of the finite element analysis relative to the exact value, p is the order of element, and k and β are constants. Applying this relationship to the two results of computations with different orders of element, the order of element for the third analysis was deduced. A computer program using this adaptive determination method for the order of element was developed and applied to several 2D elastostatic problems of various shapes. The usefulness of the method was illustrated by these application results.     

An adaptive method for the determination of the order of element for acquiring the desired accuracy in 3D elastostatic FEM analysis

An adaptive method for the determination of the order of element (or element order) was developed for the finite element analysis of 3D elastostatic problems. Here the order of element means the order of polynomial function, which interpolates the displacement distribution in the element. This method was based on acquiring the desired accuracy for each finite element. From the numerical experiments, the relationship ξ=k(1/p)^β was deduced, where ξ is the error of the result of the finite element analysis relative to the exact value, p is the order of element, and k and β are constants. Applying this relationship to the two results of computations with different orders of element, the order of element for the third computation was deduced. A computer program using this adaptive determination method for the order of element was developed and applied to several 3D elastostatic problems of various shapes. The usefulness of the method was illustrated by these application results.     

Accuracy estimation of the 3-D BEM analysis of elastostatic problems by the zooming method

A p-version zooming method for BEM is developed for 3-D elastostatic problems. The objective of this report is to develop an accuracy guarantee method for 3-D BEM analysis.First, the relations between the element degree and the analysis error of the displacement and of the traction are investigated by the numerical experiments of a thick-walled spheroid subjected to the internal or external pressure. Next, on the basis of these relations an accuracy guarantee method for the 3-D zooming method is proposed. In this method, the existence range of the analysis error of the displacement or of the traction at the point of particular interest is given by the sum of two errors. One is the analysis error of the displacement on the zooming boundary and the other is the analysis error of the traction at the point of particular interest in the zooming region. Finally, an adaptive analysis program using this method is developed and applied to the analysis of various 3-D elastostatic problems. The usefulness of this method is illustrated by these application results.     

Adaptive mesh redistribution for the boundary element in elastostatics

An adaptive mesh redistribution strategy based on the grading function method is employed to solve the two dimensional elastostatic problem with the boundary element method. Three benchmark problems, one of which involves a crack or singularity, are treated to verify the overall procedure. The results are quite satisfactory, pointing to the fact that mesh redistribution can be used effectively along with the boundary element technique.     

The Boundary Element Method with a Fast Multipole Accelerated Integration Technique for 3D Elastostatic Problems with Arbitrary Body Forces

A line integration boundary element method (LIBEM) is proposed for three-dimensional elastostatic problems with body forces. The method is a boundary-only discretization method like the traditional boundary element method (BEM), and the boundary elements created in BEM can be used directly in the proposed method for constructing the integral lines. Finally, the body forces are computed by summing one-dimensional integrals on straight lines. Background cells can be used to cut the lines into sub-lines to compute the integrals more easily and efficiently. To further reduce the computational time of LIBEM, the fast multipole method is applied to accelerate the method for large-scale computations and the details of the fast multipole line integration method for 3D elastostatic problems are given. Numerical examples are presented to demonstrate the accuracy and efficiency of the proposed method.     

A study on using hierarchical basis error estimates in anisotropic mesh adaptation for the finite element method

A common approach for generating an anisotropic mesh is the M-uniform mesh approach where an adaptive mesh is generated as a uniform one in the metric specified by a given tensor M. A key component is the determination of an appropriate metric, which is often based on some type of Hessian recovery. Recently, the use of a global hierarchical basis error estimator was proposed for the development of an anisotropic metric tensor for the adaptive finite element solution. This study discusses the use of this method for a selection of different applications. Numerical results show that the method performs well and is comparable with existing metric tensors based on Hessian recovery. Also, it can provide even better adaptation to the solution if applied to problems with gradient jumps and steep boundary layers. For the Poisson problem in a domain with a corner singularity, the new method provides meshes that are fully comparable to the theoretically optimal meshes.     

Cubic helical splines with Frenet-frame continuity

We present a scheme of interpolating a sequence of points p₀,…,pn in R³ by a piecewise-cubic-helical spline with continuous Frenet frame. Each cubic helical segment rk(ξ) connecting pk−1 and pk is constructed such that its end tangents are symmetric with respect to the displacement vector Δpk=pk−pk−1. The existence condition of such a segment rk(ξ) is formulated in terms of the configuration involving Δpk and the Frenet frame of rk(ξ) at pk−1. A Frenet-frame-continuous spline is obtained by matching the Frenet frames of sequentially constructed segments at their juncture points.     

A posteriori discontinuous finite element error estimation for two-dimensional hyperbolic problems

We analyze the discontinuous finite element errors associated with p-degree solutions for two-dimensional first-order hyperbolic problems. We show that the error on each element can be split into a dominant and less dominant component and that the leading part is O(h^p+1) and is spanned by two (p+1)-degree Radau polynomials in the x and y directions, respectively. We show that the p-degree discontinuous finite element solution is superconvergent at Radau points obtained as a tensor product of the roots of (p+1)-degree Radau polynomial. For a linear model problem, the p-degree discontinuous Galerkin solution flux exhibits a strong O(h^2p+2) local superconvergence on average at the element outflow boundary. We further establish an O(h^2p+1) global superconvergence for the solution flux at the outflow boundary of the domain. These results are used to construct simple, efficient and asymptotically correct a posteriori finite element error estimates for multi-dimensional first-order hyperbolic problems in regions where solutions are smooth.     

Numerical resolution of discontinuous Galerkin methods for time dependent wave equations

The discontinuous Galerkin (DG) method is known to provide good wave resolution properties, especially for long time simulation. In this paper, using Fourier analysis, we provide a quantitative error analysis for the semi-discrete DG method applied to time dependent linear convection equations with periodic boundary conditions. We apply the same technique to show that the error is of order k + 2 superconvergent at Radau points on each element and of order 2k + 1 superconvergent at the downwind point of each element, when using piecewise polynomials of degree k. An analysis of the fully discretized approximation is also provided. We compute the number of points per wavelength required to obtain a fixed error for several fully discrete schemes. Numerical results are provided to verify our error analysis.     

A New Nonsymmetric Discontinuous Galerkin Method for Time Dependent Convection Diffusion Equations

We propose a discontinuous Galerkin finite element method for convection diffusion equations that involves a new methodology handling the diffusion term. Test function derivative numerical flux term is introduced in the scheme formulation to balance the solution derivative numerical flux term. The scheme has a nonsymmetric structure. For general nonlinear diffusion equations, nonlinear stability of the numerical solution is obtained. Optimal kth order error estimate under energy norm is proved for linear diffusion problems with piecewise P ^k polynomial approximations. Numerical examples under one-dimensional and two-dimensional settings are carried out. Optimal (k+1)th order of accuracy with P ^k polynomial approximations is obtained on uniform and nonuniform meshes. Compared to the Baumann-Oden method and the NIPG method, the optimal convergence is recovered for even order P ^k polynomial approximations.     

Two-stage continuation algorithms for Bloch waves of Bose-Einstein condensates in optical lattices

Bloch waves of Bose-Einstein condensates (BEC) in optical lattices are extremum nonlinear eigenstates which satisfy the time-independent Gross-Pitaevskii equation (GPE). We describe an efficient Taylor predictor-Newton corrector continuation algorithm for tracing solution curves of parameter-dependent problems. Based on this algorithm, a novel two-stage continuation algorithm is developed for computing Bloch waves of 1D and 2D Bose-Einstein condensates (BEC) in optical lattices. We split the complex wave function into the sum of its real and imaginary parts. The original GPE becomes a couple of two nonlinear eigenvalue problems defined in the real domain with periodic boundary conditions. At the first stage we use the chemical potential μ as the continuation parameter. The Bloch wavenumber k(kx,ky), and the coefficient of the cubic term are treated as the second and third continuation parameters, respectively. Then we compute the Bloch bands/surfaces for the 1D/2D problem with linear counterparts. At the second stage we use μ and k/kx or ky as the continuation parameters simultaneously with two constraint conditions. The states without linear counterparts in the GPE can be obtained via states with linear counterparts. Numerical results are reported for both 1D and 2D problems.     

The simulation of 3D elastic scattering produced by thin rigid inclusions using the traction boundary element method

A mixed formulation that uses both the traction boundary element method (TBEM) and the boundary element method (BEM) is proposed to compute the three-dimensional (3D) propagation of elastic waves scattered by two-dimensional (2D) thin rigid inclusions. Although the conventional direct BEM has limitations when dealing with thin-body problems, this model overcomes that difficulty. It is formulated in the frequency domain and, taking into account the 2-1/2D configuration of the problem, can be expressed in terms of waves with varying wavenumbers in the zdirection, k_z. The elastic medium is homogeneous and unbounded and it should be noted that no restrictions are imposed on the geometry and orientation of the internal crack.     

An h-hierarchical adaptive scaled boundary finite element method for elastodynamics

A posteriori h-hierarchical adaptive scaled boundary finite element method (ASBFEM) for transient elastodynamic problems is developed. In a time step, the fields of displacement, stress, velocity and acceleration are all semi-analytical and the kinetic energy, strain energy and energy error are all semi-analytically integrated in subdomains. This makes mesh mapping very simple but accurate. Adaptive mesh refinement is also very simple because only subdomain boundaries are discretised. Two 2D examples with stress wave propagation were modelled. It is found that the degrees of freedom needed by the ASBFEM are only 5%–15% as needed by adaptive FEM for the examples.     

Noise Level Estimation in High-Dimensional Linear Models

We consider the problem of estimating the noise level σ² in a Gaussian linear model Y = Xβ+σξ, where ξ ∈ ?ⁿ is a standard discrete white Gaussian noise and β ∈ ?^p an unknown nuisance vector. It is assumed that X is a known ill-conditioned n × p matrix with n ≥ p and with large dimension p. In this situation the vector β is estimated with the help of spectral regularization of the maximum likelihood estimate, and the noise level estimate is computed with the help of adaptive (i.e., data-driven) normalization of the quadratic prediction error. For this estimate, we compute its concentration rate around the pseudo-estimate ||Y ? Xβ||²/n.     

An adaptive,multi-level method for elliptic boundary value problems

Subroutine PLTMG is a Fortran program for solving self-adjoint elliptic boundary value problems in general regions ofR ². It is based on a piecewise linear triangle finite element method, an adaptive grid refinement procedure, and a multi-level iterative method to solve the resulting sets of linear equations. In this work we describe the method and present some numerical results and comparisons.     

Accuracy estimation in the finite element analysis of transverse bending of thin flat plates

As a basic study for the establishment of an accuracy estimation method in the finite element method, this paper deals with the problems of transverse bending of thin, flat plates. From the numerical experiments for uniform mesh division, the following relation was deduced, ε ∝ (h/a)^k, k 1, where ε is the error of the computed value by the finite element method relative to the exact solution and h/a is the dimensionless mesh size. Using this relation, an accuracy estimation method, which was based on the adaptive determination of local mesh sizes from two preceding analyses by uniform mesh division, was presented.

A computer program using this accuracy estimation method was developed and applied to 28 problems with various shapes and loading conditions. The usefulness of this accuracy estimation method was illustrated by these application results.     

A priori and a posteriori error analysis for a large-scale ocean circulation finite element model

We consider the finite element solution of the stream function–vorticity formulation for a large-scale ocean circulation model. First, we study existence and uniqueness of solution for the continuous and discrete problems. Under appropriate regularity assumptions we prove that the stream function can be computed with an error of order h in H¹-seminorm. Second, we introduce and analyze an h-adaptive mesh refinement strategy to reduce the spurious oscillations and poor resolution which arise when convective terms are dominant. We propose an a posteriori anisotropic error indicator based on the recovery of the Hessian from the finite element solution, which allows us to obtain well adapted meshes. The numerical experiments show an optimal order of convergence of the adaptive scheme. Furthermore, this strategy is efficient to eliminate the oscillations around the boundary layer.     

Boundary properties of graphs for algorithmic graph problems

The notion of a boundary graph property was recently introduced as a relaxation of that of a minimal property and was applied to several problems of both algorithmic and combinatorial nature. In the present paper, we first survey recent results related to this notion and then apply it to two algorithmic graph problems: Hamiltonian cycle and vertexk-colorability. In particular, we discover the first two boundary classes for the Hamiltonian cycle problem and prove that for any k>3 there is a continuum of boundary classes for vertexk-colorability.     

A k-server problem with parallel requests and unit distances

In this paper we consider k-server problems with parallel requests where several servers can also be located on one point. We will distinguish the surplus-situation where the request can be completely fulfilled by means of the k servers and the scarcity-situation where the request cannot be completely met. We use the method of the potential function by Bartal and Grove [2] in order to prove that a corresponding Harmonic algorithm is competitive for the more general k-server problem in the case of unit distances. For this purpose we partition the set of points in relation to the online and offline servers? positions and then use detailed considerations related to sets of certain partitions.