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1. 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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2. High-order Chebyshev-based Nyström Methods for Electromagnetics

   Garza, E.; Hu, J.; Sideris, C. (August 2021, 2021 International Applied Computational Electromagnetics Society Symposium (ACES))

   Boundary element methods (BEM) have been successfully applied towards solving a broad array of complicated electromagnetic problems. Most BEM approaches rely on flat triangular discretizations and discretization via the Method of Moments (MoM) and low-order basis functions. Although more complicated from an implementation standpoint, it has been shown that high-order methods based on curvilinear patch mesh discretizations can significantly outperform low-order MoM in both accuracy and computational efficiency. In this work, we review a new high-order Nyström method based on using Chebyshev basis functions with curvilinear elements that we have recently developed, present a few scattering examples, and discuss related on-going and future work. 

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3. Untangling high-order meshes based on signed angles

   https://doi.org/10.5281/zenodo.3653412  

   Stees, Mike; Dotzel, Myra; Shontz, Suzanne M. (February 2020, Proceedings of the 28th International Meshing Roundtable)

   One challenge in the generation of high-order meshes is that mesh tangling can occur as a consequence of moving the new boundary nodes to the true curved boundary. In this paper, we propose a new optimization-based method that uses signed angles to untangle invalid second- and third-order triangular meshes. Our proposed method consists of two passes. In the first pass, we loop over each high-order interior edge node and minimize an objective function based on the signed angles of the pair of triangles that share the node. In the second pass, we loop over face nodes and move them to the mean of the high-order nodes of the triangle to which the face node belongs. 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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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. Entropy stable discontinuous Galerkin methods for the shallow water equations with subcell positivity preservation

   https://doi.org/10.1002/num.23129  

   Wu, Xinhui; Trask, Nathaniel; Chan, Jesse (July 2024, Numerical Methods for Partial Differential Equations)

   Abstract High order schemes are known to be unstable in the presence of shock discontinuities or under‐resolved solution features, and have traditionally required additional filtering, limiting, or artificial viscosity to avoid solution blow up. Entropy stable schemes address this instability by ensuring that physically relevant solutions satisfy a semi‐discrete entropy inequality independently of discretization parameters. However, additional measures must be taken to ensure that solutions satisfy physical constraints such as positivity. In this work, we present a high order entropy stable discontinuous Galerkin (ESDG) method for the nonlinear shallow water equations (SWE) on two‐dimensional (2D) triangular meshes which preserves the positivity of the water heights. The scheme combines a low order positivity preserving method with a high order entropy stable method using convex limiting. This method is entropy stable and well‐balanced for fitted meshes with continuous bathymetry profiles. 

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