Methods for connecting dissimilar three-dimensional finite element meshes

Integration of singular integrals for the Galerkin-type boundary element method in 3D elasticity

In this study, we present Balancing Domain Decomposition by Constraints (BDDC) preconditioners for three-dimensional scalar elliptic and linear elasticity problems in which the direct solution of the coarse problem is replaced by a preconditioner based on a smaller vertex-based coarse space.

This paper deals with the homogenization through Γ-convergence of weakly coercive integral energies, with the oscillating density 𝕃(x/ε)∇v:∇v, in three-dimensional elasticity. The energies are weakly coercive in the sense where the classical functional coercivity satisfied by the periodic tensor 𝕃:∫ ℝ 3 𝕃(y)∇v:∇vdy≥Λ(𝕃)∫ ℝ 3 |∇v| 2 dy,for any smooth function v with compact support in ℝ 3 , with Λ(𝕃)>0, is replaced by the relaxed condition Λ(𝕃)≥0. We prove that the homogenized tensor 𝕃 0 remains strongly elliptic, or equivalently Λ(𝕃 0 )>0, for any tensor 𝕃=𝕃(y 1 ) satisfying the pointwise inequality:𝕃(y)M:M+D:Cof(M)≥0,a.e.y∈ℝ 3 ,∀M∈ℝ 3×3 ,adding a quadratic null-Lagrangian for some matrix D∈ℝ 3×3 , and assuming the periodic functional coercivity Λ per (𝕃)>0 (using smooth test functions v with periodic gradients). However, we derive rigorously the loss of strong ellipticity for the homogenized tensor, which is based on a Γ-convergence result under the sole assumption Λ(𝕃)≥0, and on a rank-two lamination.

In this paper the sampling method for the shape reconstruction of a penetrablescatterer in three-dimensional linear elasticity is examined. We formulate thegoverning differential equations of the problem in dyadic form in order to acquirea symmetric and uniform representation for the underlying elastic fields.The corresponding far-field operator is defined in the appropriate spacesetting. We establish the interior transmission problem in the weak senseand consider the case where the nonhomogeneous boundary data aregenerated by a dyadic source point located in the interior domain. Assumingthat the inclusion has absorbing behaviour, we prove the existence anduniqueness of the weak solution of the interior transmission problem. Inthis framework the main theorem for the shape reconstruction for thetransmission case is established. As for the cases of the rigid body andthe cavity an approximate far-field equation is derived with the knowndyadic Green function term with the source point an interior point ofthe inclusion. The inversion scheme which is proposed is based on theunboundedness for the solution of an equation of the first kind. More precisely,the support of the body can be found by noting that the solution of the integralequation is not bounded as the point of the location of the fundamentalsolution approaches the boundary of the scatterer from interior points.

The three-dimensional linear dynamic elasticity problem formulation and solution for a generally asymmetric sandwich plate consisting of core and face sheets that are orthotropic, and subjected to blast loading, is presented. Laplace transforms are used to obtain ordinary differential equations in the complex Laplace space (with the variable being the through-thickness coordinate), which are subsequently solved in closed form for a simply supported plate, with the solution involving a cubic characteristic equation with complex coefficients. Subsequently, the time response is obtained by a numerical inverse Laplace transform by use of the Euler method. A realistic material and blast case is used to demonstrate the transient behavior for the displacements and face sheet/core interfacial transverse normal and shear stresses. The elasticity results are compared with the predictions of the first-order shear deformation plate theory as well as a high-order sandwich panel theory. This dynamic elasticity solution can serve as a benchmark in assessing the accuracy of sandwich-plate theories.

A relaxation method of an alternating iterative algorithm for the Cauchy problem in linear isotropic elasticity

A mixed finite element formulation is developed for the general 2D plane strain, linear isotropic gradient elasticity problem. Form II of the dipolar strain gradient theory for micro-structured solids is considered. The main variables are the double stress tensor μ and the displacement field vector u . Standard C 0 −continuous, high polynomial order hierarchical basis functions are employed for the finite element solution spaces (p-extension). The formulation is numerically validated against the standard axial tension patch test and the Mode I crack problem. The theoretical convergence rates of the uniform h - and p -extensions are confirmed using a benchmark problem where only double stresses appear. Results for the crack problem demonstrate that proper mesh refinement at areas of steep gradients ensures reproduction of the exact solution behaviour at different length scales. More specifically, the asymptotic exponents of the crack face opening displacement and the crack head true stress solutions of the Mode I crack problem are recovered. Finally, the upper bound of the true tensile normal stress near the crack tip is estimated. This upper bound is of major importance since the nature of the exact solution may change radically as we proceed from the macro- to micro-scale.

In this paper, mathematical properties of variational solutions and boundary integral equation solutions of linear elasticity problems are studied. It is first noted that a variational solution of the three-dimensional linear elasticity problem exists in the Sobolev space [H1(Ω)]3. It is then shown that a unique solution of the boundary integral equation is identical to the variational solution in [H1(Ω)]3. To represent the boundary integral equation, Green's formula in the Sobolev space is utilized on the solution domain, excluding a ball, with small radius ρ that is centered at the point where the point load is applied. By letting ρ tend to zero, it is shown that, for the linear elasticity problem, the boundary integral expression is valid for the variational solution. From this fact, a numerical approximation of the variational solution can be obtained by the boundary element method, even when a classical solution does not exist.

In this paper, an iterative algorithm based on the conjugate gradient method (CGM) in combination with the boundary element method (BEM) for obtaining stable approximate solutions to the Cauchy problem in linear elasticity is analysed. An efficient stopping criterion for the CGM proposed by Nemirovskii in 1986 is employed and in addition the accuracy of the iterative algorithm is improved by using a variable relaxation procedure. The numerical results obtained confirm that the iterative BEM produces a convergent and stable numerical solution with respect to increasing the number of boundary elements and decreasing the amount of noise added into the input data.

In this paper, we are interested in solving a Cauchy inverse problem in linear elasticity. For this, we propose a new method based on Robin conditions on the inaccessible boundary, then we study the convergence and regularizing property of the proposed algorithm. We use the finite element method for the discretization of our problem. Further, we treat the spectrum analysis of the discrete problem in order to study the convergence behavior of the proposed method in the discrete case. Finally, we present numerical results which show the efficiency and stability of the proposed approach in the presence of perturbed data. The robustness of the proposed algorithm is also performed for the anisotropic and heterogeneous cases.

Recently a refined approach to error control in finite element (FE) discretisations has been proposed, Becker and Rannacher (1995b), (1996), which uses weighted a posteriori error estimates derived via duality arguments. The conventional strategies for mesh refinement in FE models of problems from elasticity theory are mostly based on a posteriori error estimates in the energy norm. Such estimates reflect the approximation properties of the finite element ansatz by local interpolation constants while the stability properties of the continuous model enter through a global coercivity constant. However, meshes generated on the basis of such global error estimates are not appropriate in cases where the domain consists of very heterogeneous materials and for the computation of local quantities, e.g., point values or contour integrals. This deficiency is cured by using certain local norms of the dual solution directly as weights multiplying the local residuals of the computed solution. In general, these weights have to be evaluated numerically in the course of the refinement process, yielding almost optimal meshes for various kinds of error measures. This feed-back approach is developed here for primal as well as mixed FE discretisations of the fundamental problem in linear elasticity.

The paper discusses an iterative scheme for solving large-scale three-dimensional linear elasticity problems, discretized on a tensor product of two-dimensional and one-dimensional meshes. A framework is chosen of the additive AMLI method to develop a preconditioner of a black-box' type which is robust with respect to discontinuities of the problem coefficients and imposes only weak (and acceptable in practice) restrictions on the choice of the meshing procedure. The preconditioner works on a hierarchical sequence of nested finite element spaces to solve the problem with arithmetic cost, nearly proportional to the number of degrees of freedom on the finest mesh. It is particularly well suited for the case when the solution is known to be strongly varying in certain subregions of the domain and the mesh is locally prerefined there to reduce the discretization error.

Finite element method in plane Cosserat elasticity

Mathematical analysis and simulation of fixed point formulation of Cauchy problem in linear elasticity

We derive globally convergent multigrid methods for the discretized Signorini problem in linear elasticity. Special care has to be taken in the case of spatially varying normal directions. In numerical experiments for 2 and 3 space dimensions we observed similar convergence rates as for corresponding linear problems.