Adjoint‐based optimal variable stiffness mesh deformation strategy based on bi‐elliptic equations

There are many recent advances in mesh deformation methods for computational fluid dynamics simulation in deforming geometries. We present a method of constructing dynamic mesh around deforming objects by solving the bi-elliptic equation, an extension of the biharmonic equation. We show that introducing a stiffness coefficient fielda(x) in the bi-elliptic equation can enable mesh deformation for very large boundary movements. An indicator of the mesh quality is constructed as an objective function of a numerical optimization procedure to find the best stiffness coefficient field a(x). The optimization is efficiently solved using steepest descent along adjoint based, integrated Sobolev gradients. A multi-scenario optimization procedure is performed to calculate the optimal stiffness coefficient fielda � (x) for a priori unpredictable boundary movements. Simulation of fluid structure interaction problems requires solving the governing equation of the unsteady flow inside a deforming physical domain, whose geometry is determined by the deformation of the structure. In numerical methods for solving such problems, Arbitrary Lagrangian Eulerian (ALE) 1 method is one of the most popular types of approaches. It formulates the governing equation in a coordinate system that moves with the boundary, and discretizes the deforming domain with a deformable, body conforming mesh. ALE methods are capable of using anisotropic mesh to efficiently resolve boundary layers. However, these methods requires an algorithm to deform the mesh so that it conforms to the boundary movement while maintaining good mesh quality. There has been many recent advances in algorithms for computing mesh deformation in the context of computational fluid dynamics simulations in moving domains. Two of the most popular categories of such algorithms are interpolation based methods and structure analogy methods. Interpolation based methods determines the position or displacement of mesh points in the interior of the domain by interpolating the position or displacement of the mesh points on the boundary of the domain. An example of interpolation based methods is the radial basis function interpolation method 23 . Structural analogy methods include methods based on structural models of the discrete computational mesh and methods based on modeling the computational domain as a continuous elastic media. Recent developments in discrete mesh structural model includes the torsional spring method 45 . Recent developments in the continuous elasticity model include the biharmonic equation method 6 and the quasi-elliptic transformation method 7 . In this paper, we present a mesh deformation method based on continuous elasticity model. Our contribution is introducing a variable stiffness coefficient in t he biharmonic elastic model, and developing a method to efficiently obtain the optimal stiffness coefficient d istribution. We show that the variable stiffness coefficient algorithm can produce a good deformed mesh, whilethe original scheme with constant coefficient produces unacceptable mesh. To obtain a suitable stiffness coefficient, we define a simple global mesh