Approximation of the derivatives of solutions in a normalized domain for 2D solids using the PIES method
Introduction
The development of numerical methods for solving boundary value problems caused significant increase in their solvability. Special achievements in this field has the finite element method (FEM) [1], [2], [3], [4], which despite its wide possibilities, is also affected by some disadvantages. The main defect is high computational complexity of the method. It comes from the fact that the method requires generation of the element mesh by which the shape of the considered domain is modeled. This is particularly troublesome in problems with the changing shape, because it requires multiple discretization. The problem of complexity was at least partially solved with the advent of the boundary element method (BEM) [5], [6], [7], [8], because only the boundary is discretized (without the domain as in FEM). This formulation resulted in a reduction in the dimensionality of the problem by one, but we still have to build the mesh, this time the mesh of boundary elements. Moreover, in some cases (e.g. nonlinear) also the domain has to be discretized, which eliminates the main advantage of the method. A further problem of BEM is the presence of singular integrals with the various types of singularity. The determination of such integrals is difficult and troublesome, and its accuracy affects the accuracy of the final results. With singularity we deal during calculation of stresses on the boundary or in the vicinity of the boundary in linear elastic problems or stresses on the boundary and at any point of the domain in elastoplastic problems. In the latter case the issue is more complicated, because stresses are calculated at many points of the domain and many times during the iterative process as intermediate data necessary for the final solution.
The first issue described in the above paragraph, the authors have solved by developing the approach, which is an analytical modification of the boundary integral equation (BIE) [5], [6], [7], called parametric integral equations system (PIES) [9]. This modification consists in including the shape of the boundary directly in the mathematical formalism of the integral equation, which allows for flexibility in modeling. It prompts the authors to application of parametric curves (2D problems) and surfaces (3D problems), known as popular tools of computer graphics [10], [11], to the modeling of the shape. Modeling of the shape in PIES is reduced to the definition of individual segments or faces of the considered body using curves or surface patches, without classical discretization. As a result of including the shape of the boundary into the mathematical formalism of PIES we separated approximation of the shape from approximation of boundary functions, hence the possibility of modeling without classical discretization of both the boundary and the area. The proposed method, however, requires the calculation of singular integrals, the same as in BEM. In most cases, we can use the well-known and used in BEM procedures for the calculation of such integrals [5], [7]. However, there are situations in which the formulation of PIES and its characteristics cause difficulties in determining the singular integrals, and in addition used in BEM procedures are not elementary.
As a solution to the problem the authors have developed the strategy for approximation of the derivatives of solutions, which enables to obtaining e.g. stresses in continuous manner at any point of the boundary or the domain. The starting point for the development of the strategy is the fact that we can obtain stresses using basic equations of elasticity [12], in which it is necessary to know displacements and their derivatives. The displacements can be obtained by PIES using the integral identity, which is non-singular considering both elastic and plastic problems. In the case of the derivatives of displacement we can use the integral identity with possible singular integrands, or perform numerical calculation of derivatives on the basis of solutions from closely spaced points. In BEM we can also differentiate the shape functions on each element. The first method is complex, others give values at particular points. The strategy for approximation of the derivatives of solutions, which gives derivatives of any order at any points of the domain and the boundary seems to be the best solution in problems, where determination of strongly singular integrals is particularly troublesome. It should be emphasized that the proposed strategy can be applied also in BEM, but it does not guarantee the highest efficiency. About the effectiveness of the approach decides firstly the way of modeling the considered domain, the complexity of the element mesh generation and the resulting number of algebraic equations to be solved in order to achieve displacements. On the other hand, the greatest impact on the effectiveness of approximation has the arrangement of interpolation nodes and freedom in the automatic generation of such the arrangement. As is shown in the paper, PIES is a method that meets mentioned requirements.
Similar to the proposed in the paper approach is used in so-called meshless methods [13], [14], [15], [16], [17]. In this case, however, an approximation is performed locally, considering the small number of nodes due to problems with conditioning of the solved system of equations. These methods are also affected by other defects [15]. The most important, for example, is the arrangement of interpolation nodes. The way of modeling the shape developed for PIES guarantees a free and simple way of the automatic arrangement of nodes in the considered actual domain. Preliminary tests of this strategy with different node arrangements were done in the paper [18]. However, shapes considered in the paper were rather regular - mainly polygons. Whereas, it should be emphasized that if we consider a very irregular geometry we also obtain an irregular arrangement of nodes, because it is done automatically. The total lack of regularity in the distribution of interpolation nodes in the considered domain can affect the quality of the final results. For this reason, the authors decided to transpose the approximation procedure from the actual domain to the normalized domain, which is the unit square. Then the coordinates of points located in the normalized domain are mapped to the considered domain using mathematical formulas describing surface patches. Mapping is done only in order to obtain values of the function (e.g. displacement) at interpolation nodes. Finally, an approximating polynomial bases on coordinates of points from the normalized domain and displacements obtained at equivalent points in the real area. This approach allows for more accurate approximation, because regardless of the shape the arrangement of the interpolation nodes remains regular. This gives the possibility of using approximation methods impossible to use in cases with arbitrary arrangement of interpolation nodes and also affects the quality of the results.
The main aim of this paper is to develop an algorithm of the normalized strategy for approximation of derivatives of solutions in combination with advantages of the PIES method. This involves making suitable transformations from the normalized domain to the actual considered domain and vice versa. Bidirectional mapping is required in connection with the coordinates of both: interpolation nodes and those at which we want to obtain the derivatives of solutions. Transformation also require derivatives themselves, since differentiation of the approximating polynomial is done in the normalized domain. Two different methods were used for approximation, they also differ in the number and arrangement of the interpolation nodes.
Section snippets
The PIES formalism
PIES for the Navier–Lamé equation with body forces (b(y)) has been developed in [19], [20] and is presented by where and n– is the number of segments. In PIES defined in the parametric reference system, and correspond to the beginning of lth and jth segments, while and to the ends of these segments.
The first
Approximation of the derivatives of solutions in the parametric normalized domain
The solutions in the domain, formally presented by the integral identity (5) are practically calculated at each point defined by Cartesian coordinates. To obtain continuous solutions in the whole domain, as well as on the boundary we can use interpolation. For considered in the paper plane problems it is two-dimensional interpolation, which can be performed by various methods. Some of them require solving the system of equations, others only elementary arithmetic operations.
In order to verify
Required relations
Reliability and accuracy of the proposed approximation strategy is shown on the elasticity examples. The stresses were calculated. They require the knowledge of the strains, which are the partial derivatives of displacements. For this purpose, we use the constitutive relations [6] and the strain–displacement relations [6] describing isotropic, homogeneous, linear elastic body
Conclusions
Normalized approximation strategy for derivatives of solutions has been developed and applied to the analysis of 2D solids. Its elementary nature was combined with the advantages of PIES, resulting in the tool for effective use anywhere in solving boundary value problems where we have to deal with the calculation of singular integrals. The accuracy of the approach was tested on three examples of 2D elastic solids, taking into account irregular shapes. An analysis was performed considering two
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