Computational modeling and simulation of real-world problems, e.g., various automotive, aerospace, and biomedical applications, often involve geometric objects bounded by curved surfaces. Computational modeling of such objects using curvilinear high-order meshes ensures that the curved features present in the geometry are well-captured and preserved in the corresponding mesh. For a fixed level of accuracy, a high-order mesh paired with a high-order partial differential equation (PDE) solver requires fewer elements, making the mesh generation and PDE solve much less computationally expensive in total. We have developed a direct, high-order, curvilinear triangular and tetrahedral mesh generation method using an advancing front technique. Most other existing mesh generation techniques require computer-aided design (CAD) files to represent the geometric boundary, which is often unavailable for patient-specific biomedical applications. Our method requires only the element vertices and vertex connectivities to generate a high-order mesh. Thus, it can take a high-order boundary or surface mesh generated from either a CAD geometry or a set of a patient’s medical image segmentation masks as input and generate a high-order triangular or tetrahedral mesh directly from them.

In addition to a high-order mesh generator, we have also developed a finite element-based highorder mesh warping algorithm using an incompressible, hyperelastic material model. Our mesh warping algorithm takes an initial undeformed mesh in R^d d, where d = 2,3, and the corresponding boundary deformation for R^d-1. The latter could either come from mathematically-prescribed deformation fields or deformed surface meshes, e.g., deformation fields obtained from patientspecific medical images. Next, it computes the deformed mesh’s interior vertex positions using a static equilibrium condition and appropriate boundary conditions. We use our methods to generate several second-order triangular and tetrahedral static and dynamic meshes of various mechanical and anatomical models obtained from CAD models and medical images.