Fast semi-analytic computation of elastic edge singularities

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Abstract

The singularities that we consider are the characteristic non-smooth solutions of the equations of linear elasticity in piecewise homogeneous media near two-dimensional corners or three-dimensional edges. We describe here a method to compute their singularity exponents and the associated angular singular functions. We present the implementation of this method in a program whose input data are geometrical data, the elasticity coefficients of each material involved and the type of boundary conditions (Dirichlet, Neumann or mixed conditions). Our method is particularly useful with anisotropic materials and allows to “follow” the dependency of singularity exponents along a curved edge.

Introduction

In linear elasticity, problems are usually solved with industrial codes using the finite element method. However, when the elastic body has corners on its boundary, such as a polygon or a polyhedron, the solution obtained is inaccurate near the corners. The reason is that on such domains, elliptic boundary value problems admit singular solutions. Thus, special “tools” to acquire some knowledge about those solutions can be very useful.

We consider the linear equilibrium equations of an elastic material Ω, possibly heterogeneous: we suppose that Ω can be decomposed into several homogeneous parts Ωk, with k=1,…,K. Here we treat corners in two-dimensional domains and edges in three-dimensional ones. The corners and edges in question are those of Ω, of course, but also those of any of the homogeneous subdomains Ωk. The equations in Ω can be written in the general formLku=finΩk,k=1,…,Kwith boundary conditions on Ω and transmission conditions at the interfaces between the Ωk.

If Ω is two-dimensional, from the general theory (see [4], [5], or [7]), we know that, near any of its corners O, the solution can be seen as a sum of a regular part ureg and a singular one using. Except in particular cases where logarithmic terms appear, using can be expanded in the generalized formusing∼∑i=1Birνigi(θ),where (r,θ) are the polar coordinates with O as origin, Bi the stress intensity factors, νi the singularity exponents (or eigenvalues) and gi(θ) are the angular singular functions (or eigenfunctions).

The numbers νi are real or complex and characterize, along with the associated angular functions gi, the behaviour of the solution near O. They only depend on the boundary conditions, on the geometry of the subdomains Ωk, and on the material laws.

In the case of an edge, the description of the splitting into regular and singular parts is more complicated, see [3] for instance, but still involves singularity exponents νi and angular singular functions gi, possibly changing along the edge.

We are interested here in the computation of those two quantities, taking into account expressions provided by the theoretical work [1]. The core of the method is the knowledge for each ν∈C and each material index k, of an explicit solution basis of the homogeneous system without boundary conditionsLkrνg(θ)=0.Applying the boundary and transmission conditions to these bases leads to the construction of a matrix A(ν) whose determinant is called “characteristic determinant”. Singularity exponents are the roots of the equation det A(ν)=0, whose solution is obtained using Cauchy integrals.

Such a method is known and has been put into practice for isotropic materials, see [8]. But up to now, the other methods proposed for anisotropic materials are more numerical and less analytic than ours.

By a finite element approach in the one-dimensional domain Θ of the angular variable θ, Leguillon and Sanchez-Palencia in [6] construct a matrix whose eigenvalues are the singularity exponents. Yosibash [11], [12] uses a formulation of the problem based on a modified Steklov method and constructs also a matrix with a similar role, by the p-version of finite elements in a thin bi-dimensional annulus of the form {x∈R2;r0<r<r1,θ∈Θ}.

The method of Papadakis and Babuška [10] is closer to ours: they solve numerically Eq. (3) with transmission conditions and boundary conditions on one side by two initial value problems and construct a matrix Ã(ν) with the boundary conditions on the other side and then find the roots of the characteristic determinant det Ã(ν) using Cauchy integrals.

The main advantage of our method is to remain as close as possible to the exact solution since the solution bases of the equation are known almost analytically (only the roots of a symbol associated with the system Lk have to be computed). In particular, the angular singular functions gi belong to the spaces generated by the above mentioned bases. We stress again the fact that an important part of the calculation is made explicitly, which is essential considering accuracy and computation time.

In 2 Theoretical aspects: the solution bases, 3 Expression of the elasticity system, we recall from [1], [2] the determination of the solution bases of Eq. (3). In Section 4 the principles for the construction of the matrix A(ν) are explained and in Section 5 their numerical implementation is described. In 6 Computation of singularity exponents, 7 Computation of singular functions, we present the computation of the singularity exponents and of the angular singular functions. In Section 8 we treat some specific aspects of three-dimensional geometries. Finally, to show the range of application of the method and compare with the earlier works, we give in Section 9 various examples in two and three dimensions, especially for anisotropic materials, and with different boundary conditions. We draw some conclusions in Section 10.

Section snippets

Theoretical aspects: the solution bases

In this section and in the following one, we concentrate upon the solution basis of Eq. (3) for one material. Thus, we drop the material index k.

Expression of the elasticity system

Notation. In the following, we write ∂k instead of ∂xk.

The system L comes from the application of the laws of mechanics in the case of linear elasticity and depends on the classical quantities:

  • displacement field u=(u1,…,ud),

  • strain tensor εkl(u)=12(kul+luk),

  • stress tensor σij(u)=∑klaijklεkl(u), where aijkl are elasticity constants satisfying the following symmetry properties:aijkl=ajikl=aijlk=aklij.

The general equilibrium equation isLu=f,where(Lu)i=−∑jjσij(u),or, taking into account the

Characteristic determinant

We already know a solution basis of the homogeneous problem without boundary or transmission conditions. To compute the singularity exponents, we have now to take them into account.

Computation of the characteristic matrix

We describe here the computational aspects of the determination of the matrix A(ν).

Numerical aspect

Singularity exponents are the roots of the equation det A(ν)=0. To compute this determinant, we make an LU decomposition of A(ν).

The next problem is to locate the roots. For that purpose, we use Cauchy integrals over closed curves. Given holomorphic functions f and g and a closed simple contour C, Cauchy's formula is12iπCg(ν)f(ν)f(ν)dν=∑ρ∈Sg(ρ),where S is the set of the roots of f in the interior of C. Taking g(ν)=1, we can compute the number of roots inside the curve and thus isolate each

Computation of singular functions

Let ϕ̃(k)j(ν,θ) be defined as ϕ(k)j(ν,θ) for θ∈(ωk−1,ωk) and by 0 for θ∈(ω0,ωK)⧹(ωk−1,ωk). A singularity exponent ν is solution of the equation det A(ν)=0, that is to say there exists a non-zero linear combination, cf. (7),g(θ)=∑Kk=12dj=1z(k)jϕ̃(k)j(ν,θ),solution of the homogeneous problem with boundary conditions (6): any such function is an angular singular function associated with the singularity exponent ν.

Coefficients z(k)j are the components of a vector z lying in ker A(ν). In order to

Case of a three-dimensional domain

We know that in a point O lying on an edge, singularity exponents do not depend on tangential derivatives ∂z. Thus, the problem remains bi-dimensional in the plane (O;x,y). Computing singularity exponents in other points of the edge means to move the local axes along the edge and the coefficients of rigidity matrices in the local basis, because, except for isotropic material, those coefficients depend on the basis used.

Each material is assumed to be homogeneous, so coefficients are invariant

Examples

In this part, we illustrate the efficiency of the method. We compare with known results, focusing on some interesting details. At last, we propose a typical example of our own, showing what is easy to obtain in dimension 3.

All the computations have been done on a DEC Alpha workstation using double precision arithmetic. The tolerance ε had been set to 10−9.

Conclusion

In this paper, we have presented a method for the computation of singularity exponents in linear elasticity, which is especially useful in the anisotropic case. The method is based on the construction of a matrix of low dimension depending on a complex variable ν, whose determinant is 0 for a discrete set of values of this variable. These values are the exponents of singularities.

Except during the preliminary step which consists of the computation of the roots of a certain polynomial associated

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