Unified Construction of Tensor-Valued Finite Elements on Simplices Using Polytopal Templates

Alapfogalmak

This work introduces a unified method for constructing the basis functions of a wide variety of partially continuous tensor-valued finite elements on simplices using polytopal templates. The proposed approach allows for the construction of well-known elements such as Regge, Hellan-Herrmann-Johnson, Pechstein-Schöberl, Hu-Zhang, Hu-Ma-Sun, and Gopalakrishnan-Lederer-Schöberl elements.

Kivonat

The key highlights and insights of this content are:

The authors introduce a unified polytopal template method for constructing tensor-valued finite element basis functions on simplices. This method associates basis functions with the geometric polytopes (vertices, edges, faces, cells) of the reference simplex.

By using the polytopal template approach, the authors can construct a variety of well-known tensor-valued finite elements, including Regge, Hellan-Herrmann-Johnson, Pechstein-Schöberl, Hu-Zhang, Hu-Ma-Sun, and Gopalakrishnan-Lederer-Schöberl elements.

The polytopal template method allows for consistent transformations of the basis functions from the reference simplex to the physical simplex in the mesh, even for non-affine mappings. This is achieved by exploiting the association of the basis functions with the polytopes of the simplex.

The authors demonstrate that the proposed construction is independent of the underlying scalar-valued finite element basis, as long as it is H^1-conforming. This allows for higher-order, heterogeneous p-refinement, optimal complexity, and L^2-orthogonality properties to be inherited by the tensor-valued finite elements.

For non-affine mappings, the authors recommend using a hierarchical basis to ensure that the space of constants is captured by the element.

The authors provide numerical examples showcasing the application of the Hu-Zhang and Hellan-Herrmann-Johnson elements for the Reissner-Mindlin plate problem.

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Főbb Kivonatok

Formulae and transformations for simplicial tensorial finite elements via polytopal templates

by Adam Sky, Mi... : arxiv.org 09-19-2024

https://arxiv.org/pdf/2405.10402.pdf

Mélyebb kérdések

How can the proposed polytopal template method be extended to construct tensor-valued finite elements on other types of polytopes, such as hexahedra or prisms?

The polytopal template method, as introduced for simplicial tensorial finite elements, can be extended to construct tensor-valued finite elements on other types of polytopes, such as hexahedra or prisms, by adapting the underlying principles of the method to the geometric and topological characteristics of these polytopes.

Identification of Polytopes: For hexahedra, the polytopal structure includes vertices, edges, faces, and the cell itself. Each of these components can be indexed similarly to the simplicial case, allowing for the definition of multi-indices that correspond to the vertices and edges of the hexahedron.

Template Construction: The construction of tensorial template sets for hexahedra would involve defining appropriate base functions for each polytope. This includes vertex base functions that vanish on non-adjacent edges and faces, edge base functions that vanish on non-adjacent faces, and face base functions that vanish on non-adjacent edges. The tensorial bases can be constructed using dyadic products of the vectorial bases associated with these polytopes.

Transformation and Mapping: The polytopal template method relies on consistent transformations from the reference polytope to the physical polytope. For hexahedra, this would involve defining appropriate Piola transformations that account for the geometry of the hexahedron, including non-affine mappings. The mapping process must ensure that the continuity conditions required for tensor-valued functions are preserved across the interfaces of the hexahedra.

Hierarchical Basis: Similar to the approach for simplices, a hierarchical basis can be employed to ensure that the constructed tensor-valued finite elements maintain the necessary properties, such as capturing the constant space and ensuring continuity across the mesh.

By following these steps, the polytopal template method can be effectively adapted to construct tensor-valued finite elements for hexahedra and prisms, thereby broadening its applicability to a wider range of finite element analyses.

What are the potential limitations or challenges in applying the polytopal template method to construct tensor-valued finite elements for more complex partial differential equations or applications beyond linear elasticity?

While the polytopal template method offers a robust framework for constructing tensor-valued finite elements, several limitations and challenges may arise when extending its application to more complex partial differential equations (PDEs) or applications beyond linear elasticity:

Complexity of PDEs: Many complex PDEs, such as those arising in nonlinear elasticity, fluid dynamics, or multiphysics problems, may involve additional constraints or non-standard boundary conditions. The polytopal template method may need to be modified to accommodate these complexities, which could complicate the construction of the finite element spaces.

Higher Regularity Requirements: Some applications may require higher regularity of the finite element spaces than what is achievable with the standard polytopal template method. For instance, certain problems may necessitate the use of minimally regular elements, which are not guaranteed by the existing constructions. This could lead to challenges in ensuring stability and convergence of the numerical solutions.

Computational Efficiency: The computational cost associated with constructing and solving systems of equations for tensor-valued finite elements can be significant, especially in higher dimensions or for complex geometries. The polytopal template method may require efficient algorithms for assembling stiffness matrices and solving linear systems, which could be a challenge in practice.

Integration with Existing Frameworks: Integrating the polytopal template method with existing finite element frameworks or software may pose challenges, particularly if those frameworks are not designed to handle tensor-valued functions or the specific transformations required by the polytopal method.

Numerical Stability: Ensuring numerical stability when applying the polytopal template method to complex PDEs is crucial. The method must be carefully analyzed to avoid issues such as locking phenomena or spurious oscillations, which can arise in certain finite element formulations.

Addressing these challenges will require ongoing research and development to refine the polytopal template method and adapt it to the diverse needs of complex applications in computational mechanics and other fields.

Can the polytopal template approach be integrated with adaptive mesh refinement techniques to efficiently handle local singularities or complex geometries in practical simulations?

Yes, the polytopal template approach can be integrated with adaptive mesh refinement (AMR) techniques to efficiently handle local singularities or complex geometries in practical simulations. This integration can enhance the accuracy and efficiency of numerical solutions in several ways:

Local Refinement: The polytopal template method allows for the construction of tensor-valued finite elements that can be adapted locally based on the solution behavior. AMR techniques can identify regions where the solution exhibits singularities or requires higher resolution, enabling the refinement of the mesh in those specific areas while maintaining coarser elements elsewhere.

Hierarchical Basis Functions: The use of hierarchical basis functions in the polytopal template method facilitates the incorporation of AMR. As the mesh is refined, new basis functions can be introduced that maintain continuity and compatibility with the existing elements, ensuring that the overall finite element space remains well-defined.

Dynamic Mesh Adaptation: The polytopal template method can be combined with dynamic mesh adaptation strategies, where the mesh is updated during the simulation based on error estimates or solution gradients. This allows for real-time adjustments to the mesh, improving the accuracy of the solution in regions of interest without the need for a complete remeshing.

Handling Complex Geometries: The flexibility of the polytopal template method in defining tensor-valued finite elements on various polytopes makes it well-suited for complex geometries. AMR can be employed to refine the mesh around intricate features or boundaries, ensuring that the finite element representation accurately captures the geometry and the associated physical phenomena.

Error Control: By integrating AMR with the polytopal template approach, it is possible to implement error control mechanisms that guide the refinement process. This can involve assessing the local error in the tensor-valued finite element solution and refining the mesh accordingly to achieve a desired level of accuracy.

In summary, the integration of the polytopal template approach with adaptive mesh refinement techniques presents a powerful strategy for efficiently addressing local singularities and complex geometries in practical simulations, ultimately leading to more accurate and computationally efficient solutions in a variety of applications.