Designing the Optimal Shape of a Nozzle by the Method of Fundamental Solutions
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RESEARCH PAPER
Designing the Optimal Shape of a Nozzle by the Method of Fundamental Solutions Kamal Rashedi1 Received: 15 February 2020 / Accepted: 16 September 2020 Shiraz University 2020
Abstract In this paper, we propose a numerical technique based on the method of fundamental solutions (MFS) for solving a classical optimal shape design problem. The problem contains a free boundary condition which should be approximated to find the optimal domain for the solution of Laplace equation. For solving the considered optimization problem, we introduce a meshless regularization technique based on the combination of the MFS and application of the Tikhonov’s regularization method and reduce the problem to solve a system of nonlinear equations. A brief sensitivity analysis on model parameters including the position and the size of the subregion D as well the error with boundary conditions is discussed. Numerical simulations while solving several test examples are presented to show the applicability of the proposed method in obtaining satisfactory results. Keywords Elliptic equation Nozzle problem Optimal shape design Method of fundamental solutions Tikhonov regularization Mathematical Subject Classification 35A08 65N80 35R35 49K20 oX ¼ C1 [ C2 [ C3 [ C ;
1 Introduction
C1 ¼ f0g ½0; a; C2 ¼ fLg ½0; b; C3 ¼ ½0; L f0g;
Assume that the velocity of an incompressible irrotational flow at each location ðx; yÞ 2 X R2 is given by r/ where the stream function /ðx; yÞ satisfies the Laplace equation (Farahi et al. 2005) M/ ¼ 0 in
X;
with the Neumann boundary conditions 8 x 2 C1 ; > > 1; o/ < 0; x 2 C3 [ C ; ¼ on > a > : ; x 2 C2 : b such that
& Kamal Rashedi [email protected] 1
Department of Mathematics, University of Science and Technology of Mazandaran, Behshahr, Iran
ð1:1Þ
ð1:2Þ
C :¼ fðx; sðxÞÞjs 2 C½0; L; sð0Þ ¼ a; sðLÞ ¼ b; 80\x\L; 0 sðxÞ bg: ð1:3Þ The problem is defined by (Farahi et al. 2005; Mohammadi and Pironneau 2010): Z ! P : ¼ min kr/ /d k2 s:t D/ ¼ 0; oX;DX 8 D x 2 C1 ; > > 1; ð1:4Þ o/ < 0; x 2 C [ C 3 ¼ : on > a > : ; x 2 C2 : b Briefly stated, a shape C as a portion of the entire boundary oX is to be found that brings the stream function / which satisfies the two-dimensional Laplace equation in R X as well / minimizes the objective function kr/ ! 2 /d k defined over a subdomain D of the domain X (Mohammadi and Pironneau 2010). Theorem 1.1 Suppose that
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Iran J Sci Technol Trans Sci
PL C½a; b f is a piecewise linear continuous function defined on ½0; Lg:
Then, the problem given by equations (1.1)-(1.3) has at least one solution with upper bound in PL C½0; L. Proof See Farahi et al. (2005).
h
Equations (1.1–1.4) present an optimal shape design (OSD) problem which has great applications in wind tunnel or nozzle design for potential flow (Mohammadi and Pironneau 2010). For this problem, some of the boundary condition is defined over an unknown curve C oX. We address the question of simultaneous est
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