There are numerous applications for which the geometric domain moves as a function of time, e.g., flapping airplane wings, a beating heart, and clothing moving in the wind.  A dynamic sequence of meshes is required in order to capture the changing geometry. 

In the first part of the talk, I will present parallel LBWARP, a parallel log barrier-based tetrahedral mesh warping algorithm for distributed memory machines.  The algorithm is a general-purpose, geometric dynamic meshing algorithm that parallelizes the sequential LBWARP algorithm by Shontz and Vavasis.  The algorithm solves a large global system of linear equations in parallel to determine where to move the interior nodes of the mesh.  This computation is based on the representation of the initial mesh and the deformation provided by the user.  A log-barrier interior point method is used to solve several convex optimization problems to determine the representation of the initial mesh.  Sparse linear solvers for problems with multiple right-hand sides are used to solve the global systems of linear equations corresponding to multiple warping steps.  I will present several numerical examples which demonstrate the excellent scalability properties of the method. 

In the second part of the talk, I will present LBWARP2Gen, a high-order curvilinear tetrahedral mesh generation and warping algorithm.  The algorithm generates a second-order mesh by deforming a linear tetrahedral mesh into a high-order mesh based on the LBWARP method.  In particular, it first, adds a node at the midpoint of each edge; second, displaces the newly added boundary midpoints onto the curved boundary, and third, solves for the final positions of the interior nodes based on the boundary deformation (based on the representation of the initial high-order mesh and the mesh deformation).  In this case, a log-barrier interior point method is used to solve the resulting nonconvex optimization problems.  By allowing all of the boundary nodes to move, the approach can also be used to warp second-order tetrahedral meshes.  I will present numerical examples which demonstrate the success of the method in generating and warping high-order meshes. 

 Parts of this talk represent joint work by Thap Panitanarak, The Pennsylvania State University, and Michael Stees, University of Kansas.