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1. An Angular Approach to Untangling High-Order Curvilinear Triangular Meshes

   https://doi.org/10.1007/978-3-030-13992-6_18  

   Stees, Mike; Shontz, Suzanne M. (July 2019, Lecture notes in computational science and engineering)

   To achieve the full potential of high-order numerical methods for solving partial differential equations, the generation of a high-order mesh is required. One particular challenge in the generation of high-order meshes is avoiding invalid (tangled) elements that can occur as a result of moving the nodes from the low-order mesh that lie along the boundary to conform to the true curved boundary. In this paper, we propose a heuristic for correcting tangled second- and third-order meshes. For each interior edge, our method minimizes an objective function based on the unsigned angles of the pair of triangles that share the edge. We present several numerical examples in two dimensions with second- and third-order elements that demonstrate the capabilities of our method for untangling invalid meshes. 

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2. A direct high-order curvilinear triangular mesh generation method using an advancing front technique

   https://doi.org/10.1007/978-3-030-50417-5_6  

   Mohammadi, Fariba; Dangi, Shusil; Shontz, Suzanne M.; Linte, Cristian A. (June 2020, Proceedings of the 2020 International Conference on Computational Science (ICCS 2020))

   In this paper, we propose a novel method of generating high-order curvilinear triangular meshes using an advancing front approach. Our method relies on a direct approach to generate meshes on geometries with curved boundaries. Our advancing front method yields high-quality triangular elements in each iteration which omits the need for post-processing steps. We present several numerical examples of second-order curvilinear triangular meshes of patient-specific anatomical models generated using our technique on boundary meshes obtained from biomedical images. 

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3. 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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4. High-Order Cardiomyopathy Human Heart Model and Mesh Generation

   https://doi.org/10.22489/CinC.2021.139  

   Mohammadi, Fariba; Shontz, Suzanne M.; Linte, Cristian A. (December 2021, Computing in Cardiology 2021)

   Faithful, accurate, and successful cardiac biomechanics and electrophysiological simulations require patient-specific geometric models of the heart. Since the cardiac geometry consists of highly-curved boundaries, the use of high-order meshes with curved elements would ensure that the various curves and features present in the cardiac geometry are well-captured and preserved in the corresponding mesh. Most other existing mesh generation techniques require computer-aided design files to represent the geometric boundary, which are often not available for biomedical applications. Unlike such methods, our technique takes a high-order surface mesh, generated from patient medical images, as input and generates a high-order volume mesh directly from the curved surface mesh. In this paper, we use our direct high-order curvilinear tetrahedral mesh generation method [1] to generate several second-order cardiac meshes. Our meshes include the left ventricle myocardia of a healthy heart and hearts with dilated and hypertrophic cardiomyopathy. We show that our high-order cardiac meshes do not contain inverted elements and are of sufficiently high quality for use in cardiac finite element simulations. 

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5. A C 0 finite element method for the biharmonic problem with Navier boundary conditions in a polygonal domain

   https://doi.org/10.1093/imanum/drac026  

   Li, Hengguang; Yin, Peimeng; Zhang, Zhimin (July 2022, IMA Journal of Numerical Analysis)

   Abstract In this paper we study the biharmonic equation with Navier boundary conditions in a polygonal domain. In particular, we propose a method that effectively decouples the fourth-order problem as a system of Poisson equations. Our method differs from the naive mixed method that leads to two Poisson problems but only applies to convex domains; our decomposition involves a third Poisson equation to confine the solution in the correct function space, and therefore can be used in both convex and nonconvex domains. A $C^0$ finite element algorithm is in turn proposed to solve the resulting system. In addition, we derive optimal error estimates for the numerical solution on both quasi-uniform meshes and graded meshes. Numerical test results are presented to justify the theoretical findings. 

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