Abstract
The paper introduces the analytical modification of the classic boundary integral equation (BIE) for Stokes equation in 3D. The performed modification allows us to obtain separation of the approximation boundary shape from the approximation of the boundary functions. As a result, the equations, called the parametric integral equation system (PIES) with formal separation between the boundary geometry and the boundary functions, are obtained. It enables us to independently choose the most convenient methods of boundary geometry modeling depending on its complexity without any intrusion into the approximation of boundary functions and vice versa. Furthermore, we investigated the possibility of the modeling of the domains bounded by rectangular and triangular parametric Bézier patches. The boundary functions are approximated by generalized Chebyshev series. In addition, numerical techniques for solving the obtained PIES have been developed. The effectiveness of the presented strategy for boundary representation by surface patches in connection with PIES has been studied in numerical examples.
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Appendices
Appendix A: Numerical solution of PIES for 3D Stokes problem
In order to numerically solve formula (56), the pseudospectral method is used [30]. After applying (56) at the collocation points on each of the surface patches of the boundary, we obtain the system of algebraic equations from which we find values of the unknown coefficients \(\boldsymbol {u}_{r}^{(pt)} \) and \(\boldsymbol {f}_{r}^{(pt)} \) from (55). The collocation points are posed in the parametric reference (triangular or rectangular) domain \(v,w\) belongs to individual patches of the boundary. They can be placed at the roots of the Chebyshev polynomials in order to obtain the most accurate solutions, as shown by earlier studies [19]. The collocation points are defined in the parametric space domain \(v,w\) of every patch. The arrangement of collocation points in parametric space for rectangular and triangular patches are illustrated in Fig. 7.
The number of used collocation points is related directly to the number of terms in series (55), which approximate boundary functions. For rectangular patches, we assume that these points are formed in a regular structure as a Cartesian product of one dimensional grids of points defined by N rows and M columns. Their arrangement for \(N=M = 2\) and \(N=M = 3\) is shown in Fig. 7a, b. It is also possible to use more collocation points and also for a variant with \(N\ne M\)as well as introduce their different number for each surface of the boundary. For the adopted values of N and M at each patch, the series (??) are generated with terms represented by the Cartesian product of one-dimensional Chebyshev polynomials up to degree \(N-1\) and \(M-1\).
This procedure for forming collocation points may also be used in the case of triangular patches, but requires some modifications. This is due to the fact that triangular patches are related with triangular parametric space where the collocation points must be arranged as shown in Fig. 7c, d. Additionally, the number of these points and terms in series (55) is no longer defined by multiplying the number of columns and rows as for rectangular patches. This is reflected in the formula (55), through the arbitrary choice of series terms that their number corresponds to the declared number of collocation points, with ensuring symmetry-degree polynomials.
For both rectangular and triangular patches, the collocation points are arranged corresponding to the roots of the Chebyshev polynomials of the second degree This arrangement is, as shown in Fig. 7, characterized by their denser distribution on the edges. In the case of the plane of symmetry of the triangle by virtue of Chebyshev distribution further they moved away from the diagonal points of a triangle.
After writing down (56) at the collocation points \(\bar {{v}}^{(c)} ,\bar {{w}}^{(c)} \) located in the parametric domain \(v_{l-1}\! <\bar {{v}}^{(c)}\! <\!v_{l} ,w_{l-1} \!<\bar {{w}}^{(c)} \!<w_{l} \) of individual patches \(\boldsymbol {P}_{i}(l = 1,\;2,\;...,\;n)\), we obtain the system of algebraic equations in the following form
After solving (A.1), we can obtain values of the coefficients of terms (55), represented by vectors \([h_{lr} ]\) or \([g_{lj} ]\) (depending on the type of the posed boundary conditions).
Appendix B: Numerical integration
The matrices \(\boldsymbol {H}\) and \(\boldsymbol {G}\) are calculated on the basis of the following expressions requiring integration
and
An important aspect, which has to be considered during numerical solving of PIES is efficient and accurate calculation of surface integrals appearing in (B.1) and (B.2). These integrals are defined over each surface patch.
The matrix elements in (B.1) and (B.2) in the case of \(l\ne j\) requires the calculation of strongly and weak singular integrals. Integration intervals in (B.1) and (B.2) depend upon the parameterization of the patches used to model the boundary. In the case of the patches discussed in the paper, these intervals for \(F(\nu ,w)\equiv \text {\thinspace \{}\overline {\boldsymbol {{U}}}_{lr}^{\ast } \text {}(\nu ,w)\text {,}{\overline {\boldsymbol {P}}}_{lr}^{\ast } (\nu ,w)\text {\},}\) are normalized as written
where \(u = 1-v-w\).
2.1 Regular integrands
In the case of \(l\ne j\) in (B.1) and (B.2), the surface integrals are regular. To evaluate regular integrands defined over rectangular patches, standard Gauss-Legendre quadrature is used [31]. We need to transform the physical domain of integration over rectangular surface patch to the parametric domain and then to the standard interval of the Gaussian coordinates.
In the case of integrals defined over the triangular patches, their values are also computed applying Gauss-Legendre quadratures with the transformation presented in [32] and written as
where \(\xi _{i},\eta _{j} \) are Gaussian points in the \(\xi ,\eta \) directions and \(w_{i} ,w_{j} \) are the corresponding weights and s is the number of quadrature points.
2.2 Singular integrals
In the case of \(l=j\) in (B.1) and (B.2), both the collocation points as well as the numerical quadrature points lie within the same parametric domain of the same patch and kernels are strongly for matrix \([h_{lr} ]\) and weakly singular for matrix \([g_{lj} ]\), respectively. In the case of weak singularity in kernel (B.2), we isolate the singularity by introducing an additional polar transformation consisting of subdividing the local coordinate system in the parametric domain and using standard Gauss quadratures for rectangular and triangular patches. In the case of the kernel (B.1) with strong singularity, we use the method of eliminating this singularity described in [33], which are the improved version of Guiggiani’s method [34].
Appendix C: Approximation of integral identity
After solving (A.1), we obtain the solutions on the boundary. The values of the solutions in the domain \({\Omega } \) at point \(\mathrm {x}=(x_{1} ,x_{2} ,x_{3} )\) are computed by integral identity (50). After substituting approximating series (55) into (50), we give this identity in the following form
The integrand functions \(\hat {{\overline {\boldsymbol {U}}}}_{r}^{\ast } (x,\nu ,w)\) and \(\hat {{\overline {\boldsymbol {P}}}}_{r}^{\ast } (x,\nu ,w)\) are represented by (52) and (54). To determine the solution in the domain, only coefficients \(\boldsymbol {u}_{r}^{(pt)} \) and \(\boldsymbol {f}_{r}^{(pt)} \) for every surface patch which model the boundary have to be taken into account.
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Zieniuk, E., Szerszeń, K. A separation between the boundary shape and the boundary functions in the parametric integral equation system for the 3D Stokes equation. Numer Algor 80, 753–780 (2019). https://doi.org/10.1007/s11075-018-0505-3
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DOI: https://doi.org/10.1007/s11075-018-0505-3