Dependence of relative errors of the generalized () and -generalized () (, , ) solution to problem A on mesh step.
In this chapter, the two-dimensional elasticity problem with a singularity caused by the presence of a re-entrant corner on the domain boundary is considered. For this problem, the notion of the Rv-generalized solution is introduced. On the basis of the Rv-generalized solution, a scheme of the weighted finite-element method (FEM) is constructed. The proposed method provides a first-order convergence of the approximate solution to the exact one with respect to the mesh step in the W 2 , ν 1 Ω -norm. The convergence rate does not depend on the size of the angle and kind of the boundary conditions imposed on its sides. Comparative analysis of the proposed method with a classical finite-element method and with an FEM with geometric mesh refinement to the singular point is carried out.
- elasticity problem with singularity
- corner singularity
- Rv-generalized solution
- weighted finite-element method
- numerical experiments
The singularity of the solution to a boundary value problem can be caused by the degeneration of the input data (of the coefficients and right-hand sides of the equation and the boundary conditions), by the geometry of the boundary, or by the internal properties of the solution. The classic numerical methods, such as finite-difference method, finite- and boundary-element methods, have insufficient convergence rate due to singularity which has an influence on the regularity of the solution. It results in significant increase of the computational power and time required for calculation of the solution with the given accuracy. For example, the classic finite-element method allows the finding of the solution for the elasticity problem posed in a two-dimensional domain containing a re-entrant corner of
on the boundary with convergence rate O(h1/2). In this case to compute the solution with the accuracy of 10−3 requires a computational power that is one million times greater than in the case of the weighted finite-element method used for the solution of the same problem.
By using meshes refined toward the singularity point, it is possible to construct schemes of the finite-element method with the first order of the rate of convergence of the approximate solution to the exact one [1, 2, 3].
In [4, 5], for boundary value problems with strongly singular solutions for which a generalized solution could not be defined and it does not belong to the Sobolev space H1, it was proposed to define the solution as a Rv-generalized one. The existence and uniqueness of solutions as well as its coercivity and differential properties in the weighted Sobolev spaces and sets were proved [5, 6, 7, 8, 9, 10], the weighted finite-element method was built, and its convergence rate was investigated [11, 12, 13, 14, 15].
In this chapter, for the Lamé system in domains containing re-entrant corners we will state construction and investigation of the weighted FEM for determination of the Rv-generalized solution [16, 17]. Convergence rate of this method did not depend on the corner size and was equal O(h) (see , Theorem 2.1). For the elasticity problems with solutions of two types—with both singular and regular components and with singular component only—a comparative numerical analysis of the weighted finite-element method, the classic FEM, and the FEM with meshes geometrically refined toward the singularity point is performed. For the first two methods, the theoretical convergence rate estimations were confirmed. In addition, it was established that FEM with graded meshes failed on high dimensional meshes but weighted FEM stably found approximate solution with theoretical accuracy under the same computational conditions. The mentioned failure can be explained by a small size of steps of the graded mesh in a neighbourhood of the singular point. As a result, for the majority of nodes, the weighted finite-element method allows to find solution with absolute error which is by one or two orders of magnitude less than that for the FEM with graded meshes.
2. Rv-generalized solution
Let be an L-shaped domain with boundary ∂Ω containing re-entrant corner of 3π/2 with the vertex located in the point O(0,0),
Denote by a part of δ-neighbourhood of the point (0,0) laying in the . A weight function ρ(x) can be introduced that coincides with the distance to the origin in , and equals δ for .
Let be the set of functions satisfying the following conditions:
for , where k = 0,1 and c1 is a positive constant independent on k,
with the norm
where , λ = (λ1,λ2), and |λ|=λ1+λ2; λ1, λ2 are nonnegative integers, and α is a nonnegative real number.
Let be the set of functions satisfying conditions (a) and (b) with the norm
The set is defined as the closure in norm (1) of the set of infinitely differentiable and finite in Ω functions satisfying conditions (a) and (b).
One can say that if there exists a function from such that and
For the corresponding spaces and sets of vector-functions are used notations, , .
Let u = (u1,u2) be a vector-function of displacements. Assume that is a homogeneous isotropic body and the strains are small. Consider a boundary value problem for the displacement field u for the Lamé system with constant coefficients λ and μ:
Here, ε(u) is a strain tensor with components .
the bilinear and linear forms and , .
A function uv from the set is called an Rv-generalized solution to the problem (2), (3) if it satisfies boundary condition (3) almost everywhere on ∂Ω and for every v from the integral identity
holds for any fixed value of satisfying the inequality
In , for the boundary value problem (2)–(3) with homogeneous boundary conditions, existence and uniqueness of its Rv-generalized solution were established.
Let condition (4) be satisfied. Then for any there always exists parameter δ such that the problem (2)–(3) with homogeneous boundary conditions has a unique Rv-generalized solution uv in the set . In this case
where c3 is a positive constant independent of f.
Then for any , there always exists parameter δ such that the problem (2)–(3) with homogeneous boundary conditions has a unique Rv-generalized solution uv in the set .
At present, there exists a complete theory of classical solutions to boundary value problems with smooth initial data (equation coefficients, right hands of solution and boundary conditions) and with smooth enough domain boundary [19, 20, 21, 22].
On the basis of the generalized solution-wide investigations of boundary value problems with discontinuous initial data and not smooth domain boundary were performed in Sobolev and different weighted spaces [23, 24, 25, 26]. On the basis of the Galerkin method, theories of difference schemes, finite volumes, and finite-element method were developed to find approximate generalized solution .
Let us call boundary value problem a problem with strong singularity if its generalized solution could not be defined. This solution does not belong to the Sobolev space (H1), or, in other words, the Dirichlet integral of the solution diverges. In [4, 5], we suggested to define a solution to the boundary value problems with strong singularity as an Rv-generalized one in the weighted Sobolev space. The essence of this approach is in introducing weight function into the integral equality. The weight function coincides with the distance to the singular points in their neighbourhoods. The role (sense, mission) of this function is in suppressing of the solution singularity caused by the problem features and is in assuring convergence of integrals in both parts of the integral equality. Taking into account the local character of the singularity, we define weight function as the distance to each singularity point inside the disk of radius δ centered in that points, and outside these disks the weight function equals δ. An exponent of the weight function in the definition of the Rν-generalized solution as well as weighted space containing this solution depend on the spaces to which problem initial data belongs, on geometrical features of the boundary (re-entrant corners), and on changing of the boundary condition type.
In [13, 14], for the transformed system of Maxwell equations in the domain with re-entrant corner in which the solution does not depend on the space , the weighted edge-based finite-element method was developed on the basis of introducing the Rν-generalized solution. Convergence rate of this method is O(h), and it does not depend on the size of singularity as opposed to other methods [28, 29].
The proposed approach of introducing Rν-generalized solution allows to effectively find solutions not only to the boundary value problems with divergent Dirichlet integral but also to problems with weak singularity when the solution belongs to the and does not belong to the space .
3. The weighted finite-element method
A finite-element scheme for problems (2)–(3) is constructed relying on the definition of an Rν-generalized solution. For this purpose, a quasi-uniform triangulation Th of and introduction of special basis functions are constructed.
The domain is divided into a finite number of triangles K (called finite elements) with vertices Pk (k = 1,…,N), which are triangulation nodes. Denoted by —the union of all elements; here, h is the longest of their side lengths. It is required that the partition satisfies the conventional constraints imposed on triangulations . Denote by , the set of triangulation internal nodes; by , the set of nodes belonging to the ∂Ω.
Each node is associated with a function of the form
where is linear on each finite element, , is the Kronecker delta, and is a real number.
The set is defined as the linear span of the system of basis functions . Denote the corresponding vector set by . In set , one singled out the subset .
Associated with the constructed triangulation, the finite-element approximation of the displacement vector components has the form
An approximate -generalized solution to the problems (2)–(3) by the weighted finite-element method is a function such that it satisfies the boundary condition (3) in the nodes of the boundary and for arbitrary and the integral identity
holds, where .
In , it was shown that convergence rate of the approximate solution to the exact one does not depend on size of the re-entrant corner and is always equal to when weighted finite-element method is used for finding an -generalized solution to elasticity problem. The next section explains results of comparative numerical analysis for the model problems (2)–(3) of the weighted FEM using the classical finite-element method and the FEM with geometrically graded meshes of two kinds.
4. Results of numerical experiments
In the domain, is considered a Dirichlet problem for the Lamé system (2), (3) with constant coefficients and . Two kinds of vector-function were used as a solution to the problem.
Components of the solution u of the model problem (2), (3) contain only a singular component
Singularity order of u1, u2 corresponds to the size of the re-entrant corner on the domain boundary .
Solution u of the model problems (2, 3) contains both singular and regular components—regular part belongs to the
4.1. Comparative analysis of the generalized and -generalized solutions
Results of numerical experiments presented in this subsection were obtained using the code ”Proba-IV”  with regular meshes which were built by the following scheme:
Domain was divided into squares by lines parallel to coordinate axis, with distance equal to 1/N between them, where is a half of number of partitioning segments along the greater side;
Each square was subdivided into two triangles by the diagonal.
In this case, size of the mesh-step h could be computed by . Example of the regular mesh for N = 4 is presented in Figure 1.
Calculations were performed for different values of N. Optimal parameters δ, , and were obtained by the program complex . Generalized solution was determined by the integral equality (5) for .
One calculated the errors and of numerical approximation to the generalized and -generalized solutions, respectively. Problems A and B in Tables 1 and 4, respectively, present values of relative errors of the generalized solution in the norm of the Sobolev space and the -generalized one in the norm of the weighted Sobolev space with different values of . In addition, these tables contain ratios between error norms, obtained on meshes with step reducing twice. Figures 2 and 3 show the convergence rates of the generalized and -generalized solutions to the corresponding problems with the logarithmic scale. The dashed line in the figures corresponds to convergence with the rate . Tables 2 and 3 (Problem A) and Tables 5 and 6 (Problem B) give limit values: number of nodes where |e1|, |e2|, |ev,1|, and |ev,2| belong to the giving range, this number in percentage to the total number of nodes, and pictures of the absolute error distribution in the domain Ω.
4.1.1. Problem A
4.1.2. Problem B
4.2. FEM with graded mesh: comparative analysis
This subsection presents results of error analysis for finding generalized solution to the problems A and B by the FEM with graded meshes of two kinds (for detailed information about graded meshes, see [2, 33, 34]).
Mesh I. This partitioning was built by the following scheme
In the domain , for a given N, regular mesh was constructed as described in section 4.1.
Level was determined for each node. Here, () are initial node coordinates on the regular mesh, means integer part.
New coordinates of nodes of the graded mesh are calculated by the formula ().
Mesh II. Constructing process for this mesh differs from the one described earlier in the level-calculating mode. Here, . In this case, new coordinates are determined only for nodes with .
Calculations were performed for different values of and . For each node, one calculated the errors and of the approximate generalized solutions , obtained on meshes I and II, respectively. The values of relative errors of the generalized solution to the problems A and B in the norm of the Sobolev space for different values of and for mesh I are presented in Tables 7 and 10, respectively, and for mesh II are presented in Tables 8 and 11, respectively. In addition, these tables contain ratios between error norms and between mesh steps obtained with nodes number increasing four times. Figures 4 and 5 show the convergence rates of the generalized solutions to the corresponding problems for meshes I and II with the logarithmic scale. Dashed line in the figures corresponds to convergence with the rate O(h) as in paragraph 1. Besides, for the problems A and B, Tables 9 and 12, respectively, contain limit values for the following data: number of nodes where , belong to the giving range, this number in percentage to the total number of nodes, and pictures of the absolute error distribution in the domain .
4.2.1. Problem A
4.2.2. Problem B
Presented numerical results have demonstrated that:
An approximate -generalized solution to the problem (2)–(4) converges to the exact one with the rate in the norm of the set in contrast with the generalized solution, which converges with the rate for the classical FEM;
FEM with graded meshes fails on high-dimensional grids because of the small mesh size near the singular point, but the weighted FEM stably allows to find approximate solution with the accuracy under the same computational conditions;
For the approximate -generalized solution obtained by the weighted finite-element method, an absolute error value is by one or two orders of magnitude less than the approximate generalized one obtained by the FEM or by the FEM with graded meshes; this holds for the overwhelming majority of nodes.