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Solving biharmonic equation as an optimal control problem using localized radial basis functions collocation method

机译：用局部径向基函数搭配法求解双调和方程作为最优控制问题

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摘要

Solving fourth or higher order differential equations using localized numerical methods as the finite element, the finite difference and the localized radial basis functions (LRBFs), show difficulties to get accurate results. So many authors adopted iterative methods to deal with biharmonic equation by decoupling it into two Poissons problems. In this paper we investigate the formulation of a mixed fourth order boundary value problem as an optimal control one. Then, we establish a new iterative method by coupling an optimization iterative scheme and localized radial basis functions meshless collocation method to deal with the numerical solution of such problem. To transform the problem into an optimal control one, we firstly construct the constraints functions by splitting the biharmonic equation into two coupled Laplace equations. The Neumann boundary condition is used as energy-like error functional to be minimized. Theoretical analysis of the existence and uniqueness of the solution of such formulated optimization problem and its equivalence to the initial biharmonic problem are also demonstrated. Finally we show the effectiveness of the proposed method by solving problems in both convex and non-convex regular and irregular domain.
机译：使用局部数值方法作为有限元，有限差分和局部径向基函数（LRBF）求解四阶或更高阶微分方程，显示出难以获得准确结果的困难。因此，许多作者采用迭代方法将双谐波方程解耦为两个Poissons问题。在本文中，我们研究了混合四阶边值问题作为最优控制问题的表述。然后，通过结合优化迭代方案和局部径向基函数无网格配置方法，建立了一种新的迭代方法，以解决该问题的数值解。为了将问题转化为最优控制，我们首先通过将双谐波方程分解为两个耦合的拉普拉斯方程来构造约束函数。诺伊曼边界条件被用作要最小化的类能量误差函数。还证明了这种优化问题解的存在性和唯一性及其与初始双调和问题的等价性的理论分析。最后，我们通过解决凸和非凸规则与不规则域中的问题，证明了该方法的有效性。

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