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1. Proceedings of the 2025 SIAM International Meshing Roundtable; Parallelization of the Finite Element-based Mesh Warping Algorithm Using Hybrid Parallel Programming

   https://doi.org/10.1137/1.9781611978575.12  

   Haque, Abir; Shontz, Suzanne M (March 2025, Society for Industrial and Applied Mathematics)

   Si, Hang; Shepherd, Kendrick M; Zhang, Yongjie Jessica (Ed.)

   Warping large volume meshes has applications in biomechanics, aerodynamics, image processing, and cardiology. However, warping large, real-world meshes is computationally expensive. Existing parallel implementations of mesh warping algorithms do not take advantage of shared-memory and one-sided communication features available in the MPI-3 standard. We describe our parallelization of the finite element-based mesh warping algorithm for tetrahedral meshes. Our implementation is portable across shared and distributed memory architectures, as it takes advantage of shared memory and one-sided communication to precompute neighbor lists in parallel. We then deform a mesh by solving a Poisson boundary value problem and the resulting linear system, which has multiple right-hand sides, in parallel. Our results demonstrate excellent efficiency and strong scalability on up to 32 cores on a single node. Furthermore, we show a 33.9% increase in speedup with 256 cores distributed uniformly across 64 nodes versus our largest single node speedup while observing sublinear speedups overall. 

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2. A direct method for generating quadratic curvilinear tetrahedral meshes using an advancing front approach

   https://doi.org/10.5281/zenodo.5559211  

   Mohammadi, Fariba; Shontz, Suzanne M. (October 2021, Proc. of the 29th International Meshing Roundtable)

   Computational modeling and simulation of real-world problems, e.g., various applications in the automotive, aerospace, and biomedical industries, often involve geometric objects which are bounded by curved surfaces. The geometric modeling of such objects can be performed via high-order meshes. Such a mesh, when paired with a high-order partial differential equation (PDE) solver, can realize more accurate solution results with a decreased number of mesh elements (in comparison to a low-order mesh). There are several types of high-order mesh generation approaches, such as direct methods, a posteriori methods, and isogeometric analysis (IGA)-based spline modeling approaches. In this paper, we propose a direct, high-order, curvilinear tetrahedral mesh generation method using an advancing front technique. After generating the mesh, we apply mesh optimization to improve the quality and to take advantage of the degrees of freedom available in the initially straight-sided quadratic elements. Our method aims to generate high-quality tetrahedral mesh elements from various types of boundary representations including the cases where no computer-aided design files are available. Such a method is essential, for example, for generating meshes for various biomedical models where the boundary representation is obtained from medical images instead of CAD files. We present several numerical examples of second-order tetrahedral meshes generated using our method based on input triangular surface meshes. 

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3. A Moving Mesh Adaption Method By Optimal Transport

   An, Dongsheng; Lei, Na; Zhao, Tong; Si, Hang; Gu, Xianfeng (October 2021, Proceedings of International Meshing Roundtable)

   null (Ed.)

   Optimal transportation finds the most economical way to transport one probability measure to another, and it plays an important role in geometric modeling and processing. In this paper, we propose a moving mesh method to generate adaptive meshes by optimal transport. Given an initial mesh and a scalar density function defined on the mesh domain, our method will redistribute the mesh nodes such that they are adapted to the density function. Based on the Brenier theorem, solving an optimal transportation problem is reduced to solving a Monge-Amp\`ere equation, which is difficult to compute due to the high non-linearity. On the other hand, the optimal transportation problem is equivalent to the Alexandrov problem, which can finally induce an effective variational algorithm. Experiments show that our proposed method finds the adaptive mesh quickly and efficiently. 

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4. A fully-coupled framework for solving Cahn-Hilliard Navier-Stokes equations: Second-order, energy-stable numerical methods on adaptive octree based meshes

   https://doi.org/https://doi.org/10.1016/j.cpc.2022.108501  

   Makrand A. Khanwale; Kumar Saurabh; Milinda Fernando; Victor M.Calo; Hari Sundar; James A.Rossmanith; Baskar Ganapathysubramanian (November 2022, Computer physics communications)

   We present a fully-coupled, implicit-in-time framework for solving a thermodynamically-consistent Cahn-Hilliard Navier-Stokes system that models two-phase flows. In this work, we extend the block iterative method presented in Khanwale et al. [Simulating two-phase flows with thermodynamically consistent energy stable Cahn-Hilliard Navier-Stokes equations on parallel adaptive octree based meshes, J. Comput. Phys. (2020)], to a fully-coupled, provably second-order accurate scheme in time, while maintaining energy-stability. The new method requires fewer matrix assemblies in each Newton iteration resulting in faster solution time. The method is based on a fully-implicit Crank-Nicolson scheme in time and a pressure stabilization for an equal order Galerkin formulation. That is, we use a conforming continuous Galerkin (cG) finite element method in space equipped with a residual-based variational multiscale (RBVMS) procedure to stabilize the pressure. We deploy this approach on a massively parallel numerical implementation using parallel octree-based adaptive meshes. We present comprehensive numerical experiments showing detailed comparisons with results from the literature for canonical cases, including the single bubble rise, Rayleigh-Taylor instability, and lid-driven cavity flow problems. We analyze in detail the scaling of our numerical implementation. 

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5. Decoupling Simulation Accuracy from Mesh Quality

   Schneider, Teseo; Hu, Yixin; Dumas, Jeremie; Gao, Xifeng; Panozzo, Daniele; Zorin, Denis (December 2018, ACM transactions on graphics)

   For a given PDE problem, three main factors affect the accuracy of FEM solutions: basis order, mesh resolution, and mesh element quality. The first two factors are easy to control, while controlling element shape quality is a challenge, with fundamental limitations on what can be achieved. We propose to use p-refinement (increasing element degree) to decouple the approximation error of the finite element method from the domain mesh quality for elliptic PDEs. Our technique produces an accurate solution even on meshes with badly shaped elements, with a slightly higher running time due to the higher cost of high-order elements. We demonstrate that it is able to automatically adapt the basis to badly shaped elements, ensuring an error consistent with high-quality meshing, without any per-mesh parameter tuning. Our construction reduces to traditional fixed-degree FEM methods on high-quality meshes with identical performance. Our construction decreases the burden on meshing algorithms, reducing the need for often expensive mesh optimization and automatically compensates for badly shaped elements, which are present due to boundary con- straints or limitations of current meshing methods. By tackling mesh gen- eration and finite element simulation jointly, we obtain a pipeline that is both more efficient and more robust than combinations of existing state of the art meshing and FEM algorithms. 

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