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1. A Large-Scale Comparison of Tetrahedral and Hexahedral Elements for Solving Elliptic PDEs with the Finite Element Method

   https://doi.org/10.1145/3508372  

   Schneider, Teseo; Hu, Yixin; Gao, Xifeng; Dumas, Jérémie; Zorin, Denis; Panozzo, Daniele (June 2022, ACM Transactions on Graphics)

   The Finite Element Method (FEM) is widely used to solve discrete Partial Differential Equations (PDEs) in engineering and graphics applications. The popularity of FEM led to the development of a large family of variants, most of which require a tetrahedral or hexahedral mesh to construct the basis. While the theoretical properties of FEM basis (such as convergence rate, stability, etc.) are well understood under specific assumptions on the mesh quality, their practical performance, influenced both by the choice of the basis construction and quality of mesh generation, have not been systematically documented for large collections of automatically meshed 3D geometries. We introduce a set of benchmark problems involving most commonly solved elliptic PDEs, starting from simple cases with an analytical solution, moving to commonly used test problem setups, and using manufactured solutions for thousands of real-world, automatically meshed geometries. For all these cases, we use state-of-the-art meshing tools to create both tetrahedral and hexahedral meshes, and compare the performance of different element types for common elliptic PDEs. The goal of this benchmark is to enable comparison of complete FEM pipelines, from mesh generation to algebraic solver, and exploration of relative impact of different factors on the overall system performance. As a specific application of our geometry and benchmark dataset, we explore the question of relative advantages of unstructured (triangular/ tetrahedral) and structured (quadrilateral/hexahedral) discretizations. We observe that for Lagrange-type elements, while linear tetrahedral elements perform poorly, quadratic tetrahedral elements perform equally well or outperform hexahedral elements for our set of problems and currently available mesh generation algorithms. This observation suggests that for common problems in structural analysis, thermal analysis, and low Reynolds number flows, high-quality results can be obtained with unstructured tetrahedral meshes, which can be created robustly and automatically. We release the description of the benchmark problems, meshes, and reference implementation of our testing infrastructure to enable statistically significant comparisons between different FE methods, which we hope will be helpful in the development of new meshing and FEA techniques. 

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2. Construction of Scalar and Vector Finite Element Families on Polygonal and Polyhedral Meshes

   https://doi.org/10.1515/cmam-2016-0019  

   Gillette, Andrew; Rand, Alexander; Bajaj, Chandrajit (January 2016, Computational Methods in Applied Mathematics)

   Abstract We combine theoretical results from polytope domain meshing, generalized barycentric coordinates, and finite element exterior calculus to construct scalar- and vector-valued basis functions for conforming finite element methods on generic convex polytope meshes in dimensions 2 and 3. Our construction recovers well-known bases for the lowest order Nédélec, Raviart–Thomas, and Brezzi–Douglas–Marini elements on simplicial meshes and generalizes the notion of Whitney forms to non-simplicial convex polygons and polyhedra. We show that our basis functions lie in the correct function space with regards to global continuity and that they reproduce the requisite polynomial differential forms described by finite element exterior calculus. We present a method to count the number of basis functions required to ensure these two key properties. 

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3. Artifact-Based Rendering: Harnessing Natural and Traditional Visual Media for More Expressive and Engaging 3D Visualizations

   https://doi.org/10.1109/TVCG.2019.2934260  

   Johnson, Seth; Samsel, Francesca; Abram, Gregory; Olson, Daniel; Solis, Andrew J.; Herman, Bridger; Wolfram, Phillip J.; Lenglet, Christophe; Keefe, Daniel F. (August 2019, IEEE Transactions on Visualization and Computer Graphics)

   We introduce Artifact-Based Rendering (ABR), a framework of tools, algorithms, and processes that makes it possible to produce real, data-driven 3D scientific visualizations with a visual language derived entirely from colors, lines, textures, and forms created using traditional physical media or found in nature. A theory and process for ABR is presented to address three current needs: (i) designing better visualizations by making it possible for non-programmers to rapidly design and critique many alternative data-to-visual mappings; (ii) expanding the visual vocabulary used in scientific visualizations to depict increasingly complex multivariate data; (iii) bringing a more engaging, natural, and human-relatable handcrafted aesthetic to data visualization. New tools and algorithms to support ABR include front-end applets for constructing artifact-based colormaps, optimizing 3D scanned meshes for use in data visualization, and synthesizing textures from artifacts. These are complemented by an interactive rendering engine with custom algorithms and interfaces that demonstrate multiple new visual styles for depicting point, line, surface, and volume data. A within-the-research-team design study provides early evidence of the shift in visualization design processes that ABR is believed to enable when compared to traditional scientific visualization systems. Qualitative user feedback on applications to climate science and brain imaging support the utility of ABR for scientific discovery and public communication. 

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4. 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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5. Robust automatic hexahedral cartilage meshing framework enables population-based computational studies of the knee

   https://doi.org/10.3389/fbioe.2022.1059003  

   Gibbons, Kalin D.; Malbouby, Vahid; Alvarez, Oliver; Fitzpatrick, Clare K. (December 2022, Frontiers in Bioengineering and Biotechnology)

   Osteoarthritis of the knee is increasingly prevalent as our population ages, representing an increasing financial burden, and severely impacting quality of life. The invasiveness of in vivo procedures and the high cost of cadaveric studies has left computational tools uniquely suited to study knee biomechanics. Developments in deep learning have great potential for efficiently generating large-scale datasets to enable researchers to perform population-sized investigations, but the time and effort associated with producing robust hexahedral meshes has been a limiting factor in expanding finite element studies to encompass a population. Here we developed a fully automated pipeline capable of taking magnetic resonance knee images and producing a working finite element simulation. We trained an encoder-decoder convolutional neural network to perform semantic image segmentation on the Imorphics dataset provided through the Osteoarthritis Initiative. The Imorphics dataset contained 176 image sequences with varying levels of cartilage degradation. Starting from an open-source swept-extrusion meshing algorithm, we further developed this algorithm until it could produce high quality meshes for every sequence and we applied a template-mapping procedure to automatically place soft-tissue attachment points. The meshing algorithm produced simulation-ready meshes for all 176 sequences, regardless of the use of provided (manually reconstructed) or predicted (automatically generated) segmentation labels. The average time to mesh all bones and cartilage tissues was less than 2 min per knee on an AMD Ryzen 5600X processor, using a parallel pool of three workers for bone meshing, followed by a pool of four workers meshing the four cartilage tissues. Of the 176 sequences with provided segmentation labels, 86% of the resulting meshes completed a simulated flexion-extension activity. We used a reserved testing dataset of 28 sequences unseen during network training to produce simulations derived from predicted labels. We compared tibiofemoral contact mechanics between manual and automated reconstructions for the 24 pairs of successful finite element simulations from this set, resulting in mean root-mean-squared differences under 20% of their respective min-max norms. In combination with further advancements in deep learning, this framework represents a feasible pipeline to produce population sized finite element studies of the natural knee from subject-specific models. 

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