三维Helmholtz方程的等几何边界元法和伴随变量法形状优化
A shape optimisation with the isogeometric boundary element method and adjoint variable method for the three-dimensional Helmholtz equation
摘要：
本文提出了一个形状优化系统，设计的声学硬物体在三维开放空间的形状。边界元法（BEM）适合于分析这样的外场。然而，传统的边界元法，这是基于分段多项式形状和插值函数，可以需要许多设计变量，因为它们通常被选为底层边界元网格的节点的一部分。此外，它是不容易的传统方法来计算的表面上的声压的梯度，这是必要的计算我们感兴趣的形状导数，一个给定的对象。为了克服这些问题，我们采用等几何边界元法（IGBEM），这是在我们以前的工作。利用IGBEM，我们可以通过目标对象的NURBS曲面的控制点来设计曲面的形状。我们通过伴随变量法（AVM）将IGBEM与非线性规划软件相结合，由此产生的伴随边值问题也可以通过IGBEM进行轻微修改来求解。数值验证和演示验证了我们的形状优化框架。
英文摘要：
This paper presents a shape optimisation system to design the shape of an acoustically-hard object in the three-dimensional open space. Boundary element method (BEM) is suitable to analyse such an exterior field. However, the conventional BEM, which is based on piecewise polynomial shape and interpolation functions, can require many design variables because they are usually chosen as a part of the nodes of the underlying boundary element mesh. In addition, it is not easy for the conventional method to compute the gradient of the sound pressure on the surface, which is necessary to compute the shape derivative of our interest, of a given object. To overcome these issues, we employ the isogeometric boundary element method (IGBEM), which was developed in our previous work. With using the IGBEM, we can design the shape of surfaces through control points of the NURBS surfaces of the target object. We integrate the IGBEM with the nonlinear programming software through the adjoint variable method (AVM), where the resulting adjoint boundary value problem can be also solved by the IGBEM with a slight modification. The numerical verification and demonstration validate our shape optimisation framework.
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