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Best Curved Adapted Meshes for Space-Time Flow Simulations

Periodic Reporting for period 2 - Tesseract (Best Curved Adapted Meshes for Space-Time Flow Simulations)

Reporting period: 2018-10-01 to 2020-03-31

The accuracy obtained in wind tunnel aerodynamic and aero-acoustic measurements is extremely demanding and it still challenges our current simulation technologies. It mainly challenges the capabilities of current mesh generation technologies used in flow simulations. The ground-breaking TESSERACT project addresses the challenge of studying how to generate computational meshes that enable the ability to obtain computer flow simulations that beat the predictive capabilities of the wind tunnel experiments for a fixed accuracy, cost, and time scale. These important challenges correspond to capabilities that have been considered essential to fulfil the European strategic goals of future transportation. The main objective is to generate optimal quality curved adapted meshes for space-time flow simulations by addressing the following ambitious and beyond the state of the art 4-dimensional meshing research objectives: curved geometry representation and approximation, mesh quality measures, adapted mesh resolution, and space-time flow simulation. This is a high risk project since it tackles meshing objectives in 4D while lower dimension versions of these issues have not yet been fully solved. However, providing the foundations and the methods to improve current space-time meshing algorithms will suppose a high gain in the field of computational and aerospace engineering. This is so since in the near future, it will be of major importance to conduct accurate, robust, and efficient parallel in space-time adapted flow simulations that exploit the computational power of the exascale super-computing facilities to come. To enhance the feasibility of the project, the scientific approach considers different novel approaches to reach the same objectives and therefore, bear in mind the high-risk / high-gain nature of this 4D meshing project.
- Defining a regularized shape distortion (quality) measure for curved high-order 2D elements on a Riemannian plane. The defined distortion (quality) is suitable to check the validity and the quality of straight-sided and curved elements on Riemannian planes determined by constant and point-wise varying metrics. The examples illustrate that the distortion can be minimized to curve (deform) the elements of a given high-order (linear) mesh and try to match with curved (linear) elements the point-wise alignment and stretching of an analytic target metric tensor.

- Local bisection for refinement of 4D simplicial meshes. We have proposed a recursive refine to conformity procedure in two stages. The method ensures that any 4D unstructured mesh can be conformingly refined with bounded minimum shape quality. After successive refinement the mesh quality does not degenerate. Moreover, we can refine a 4D unstructured mesh and a space-time mesh (3D + 1D) representation of a moving object.

- Automatically imposing incremental boundary displacements for valid mesh morphing and curving. Our method seeks a diffeomorphism that transforms an initial domain to a final domain by only prescribing the boundary displacement. Our approach explicitly penalizes the appearance of non-invertible mappings and therefore, we do not need to equip our discrete implementation with untangling capabilities. Our mesh morphing method is suitable for large displacements and rotations of meshes with non-uniform sizing, and mesh curving of highly stretched high-order meshes.

- Imposing boundary conditions to match a CAD virtual geometry for the mesh curving problem. Our method has the unique capability to allow curved elements to span and slide on top of several CAD entities during the mesh curving process. The main advantage is that small angles or small patches of the CAD model do not compromise the topology, quality and size of the boundary elements. The method is suitable to curve meshes featuring non-uniform isotropic and highly stretched elements while matching a given virtual geometry.

- A parallel distributed approach to refine a mesh while preserving the curvature of a target geometry is presented. The result of our approach is a distributed finer linear or high-order mesh featuring improved geometric accuracy. The qualitative results show that for steady state flow solutions our parallel subdivision approach mitigates the artificial artifacts that might appear with standard straight-sided subdivision methods. We also check the parallel performance of the implementation by performing a weak scalability test.

- A a simple tool to visualize 4D unstructured pentatopic meshes. The results show that the method is suitable to visually explore 4D unstructured meshes. This capability has facilitated devising our 4D bisection method, and thus, we think it might be useful when devising new 4D meshing methods. Furthermore, it allows visualizing 4D scalar fields, which is a crucial feature for our space-time applications.

- A fast curving method based on hierarchical subdivision and blending. There is no need for underlying target geometry, it is only needed a straight-edged mesh with boundary entities marked to characterize the geometry features, and a list of features to recast. The method features a unique sharp-to-smooth modeling capability not fully available in standard CAD packages. We conclude that the method is well-suited to curve large quadratic and quartic meshes in low-memory configurations.

- An optimization method to adapt straight-edged and curved piece-wise polynomial meshes to the stretching and alignment of a target metric. To compare both globalization approaches, we derive two specific-purpose implementations of Newton's method equipped with backtracking line search (BLS) and restricted trust region (RTR). We have been able to improve the inexact Newton implementation, with both globalization methods, to reduce one order of magnitude the total number of sparse matrix-vector products

- To generate large-scale meshes with highly stretched elements for complex geometries, we propose a distributed parallel Newton-GMRES penalty solver. We show that an additive Schwarz domain decomposition is faster and more energy efficient than an algebraic multigrid pre-conditioner. Furthermore, we propose a novel p-continuation technique equipped with: an early termination criterion, and an estimation of the initial penalty parameter. We conclude that this continuation can reduce four times (eight times) the wall clock time (energy per core) required to curve a whole boundary layer quartic mesh using the chosen domain decomposition.
All the results achieved so far have been peer-reviewed and therefore, the following list represents our progress beyond the state of the art:

- Defining a regularized shape distortion (quality) measure for curved high-order 2D elements on a Riemannian plane.
- Local bisection for refinement of 4D simplicial meshes.
- Automatically imposing incremental boundary displacements for valid mesh morphing and curving.
- Imposing boundary conditions to match a CAD virtual geometry for the mesh curving problem.
- Parallel curved mesh subdivision.
- Slice-based visualization of unstructured 4D simplicial meshes and fields.
- Subdivided linear and curved meshes preserving features of a linear mesh model.
- Specific-purpose globalized non-linear solvers for anisotropic optimization of curved meshes.
- Pre-conditioning and continuation for parallel distributed mesh curving.