[Paper Review] A variational approach to high-order r-adaptation

Julian Marcon, Michael G. Turner+3 more|arXiv (Cornell University)|Sep 18, 2017

Seismic Imaging and Inversion Techniques6 references4 citations

TL;DR

This paper proposes a variational framework for high-order r-adaptation by manipulating target elements via metric tensors to guide node relocation, minimizing element distortion. The method successfully achieves anisotropic, high-order mesh adaptation with improved resolution near curved features while preserving mesh connectivity and degrees of freedom.

ABSTRACT

A variational framework, initially developed for high-order mesh optimisation, is being extended for r-adaptation. The method is based on the minimisation of a functional of the mesh deformation. To achieve adaptation, elements of the initial mesh are manipulated using metric tensors to obtain target elements. The nonlinear optimisation in turns adapts the final high-order mesh to best fit the description of the target elements by minimising the element distortion. Encouraging preliminary results prove that the method behaves well and can be used in the future for more extensive work which shall include the use of error indicators from CFD simulations.

Motivation & Objective

• To extend a variational mesh optimization framework to enable r-adaptation in high-order finite element methods.
• To maintain mesh connectivity and degrees of freedom while concentrating resolution in regions of interest.
• To demonstrate the feasibility of using metric tensors to define target element shapes for adaptive meshing.
• To provide a foundation for future integration with error indicators from CFD simulations.
• To enable efficient, node-based adaptation for unsteady flow simulations without costly remeshing.

Proposed method

• The method uses a variational formulation based on hyperelastic deformation energy to minimize mesh distortion.
• Target elements are defined via metric tensors that encode desired element shape and size, enabling isotropic or anisotropic adaptation.
• The Jacobian of the target mapping is modified through linear transformations: rotation, scaling, and re-rotation to achieve directional refinement.
• A Gaussian distribution is used to define a spatially varying scaling factor in the radial direction, creating targeted refinement around a circular region.
• The algorithm relocates nodes by minimizing the deformation energy functional between the current mesh and the target metric-defined elements.
• Each element's target is computed at its barycenter using a single metric tensor, resulting in linear target elements that are then curved by the optimization process.

Experimental results

Research questions

• RQ1 Can a variational mesh optimization framework be extended to support r-adaptation in high-order meshes?
• RQ2 How can metric tensors be used to define anisotropic target element shapes for targeted mesh refinement?
• RQ3 What is the behavior of the method in terms of element quality and resolution concentration in curved regions?
• RQ4 Can the method preserve mesh connectivity and degrees of freedom while achieving high-resolution adaptation?
• RQ5 How can this approach be generalized to use error indicators from CFD simulations in future work?

Key findings

• The method successfully achieves anisotropic refinement along a unit-diameter circular circumference, with elements shrunk in the radial direction only.
• The adapted mesh shows significant resolution concentration near the target region, with coarsening observed in other areas due to node redistribution.
• The method preserves element connectivity and maintains a constant number of degrees of freedom throughout adaptation.
• The algorithm produces highly curved elements near the target region, consistent with the expected behavior of high-order mesh adaptation.
• The use of a single metric tensor per element at the barycenter results in linear target elements, which are then deformed into curvilinear shapes by the optimization process.
• Preliminary results confirm the method's feasibility and robustness, with strong potential for integration with error indicators in future work.