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Best Curved Adapted Meshes for Space-Time Flow Simulations

Periodic Reporting for period 3 - Tesseract (Best Curved Adapted Meshes for Space-Time Flow Simulations)

Reporting period: 2020-04-01 to 2021-09-30

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.

- 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 enhance the geometric accuracy of piece-wise polynomial approximations of target compound geometries, we propose an adaptive procedure based on a two-times differentiable disparity functional. We show numerical evidence that the obtained curved high-order meshes super-converge to the target geometry. The convergence exponent is two times the polynomial degree. This approach approximates the target geometry with the same convergence rate as a standard interpolation method but with an almost two times smaller polynomial degree.

- We propose an adaptive penalty and degree continuation for parallel non-linear pre-conditioned mesh curving. The solver features reduced memory footprint, computational cost, and energy consumption. The method aims to tackle large-scale curved meshing challenges for high-fidelity simulation such as the ones posed in the Hi-Lift Workshop.

- A method for mesh-based curved geometry modeling. The technique reconstructs a curved geometry representation that preserves the sharp initial features from a linear mesh model and smoothes the indicated ones. Given an arbitrary degree and nodal distribution, the method provides a piece-wise polynomial representation of complex geometries while preserving the simulation intent.
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.
- Specific-purpose globalized non-linear solvers for anisotropic optimization of curved meshes.
- Parallel mesh curving on complex virtual geometry.
- Measuring and improving the geometric accuracy of piece-wise polynomial meshes
- Efficient solver to optimize the geometric accuracy
- Sub-optimal nD bisection for local refinement of unstructured conformal meshes
- Optimal 3D bisection for local refinement of unstructured conformal meshes
- High-order metric interpolation for curved adaption
- Interpolation of subdivision features for curved geometry modeling
- Sub-optimal interpolation points on simplices
- Morphing and topological modification for space-time meshing
Conformal bisection for 4D simplicial meshes: space-time refinement
Imposing weak boundary conditions: curved mesh of a propeller
Conformal bisection for 4D simplicial meshes: unstructured mesh
Conformal bisection for 4D simplicial meshes: slices are like the frames of a 3D movie
Optimizing stretching and alingment for curved meshes: better matching with fewer elements
Parallel mesh curved subidivision: weak scaling and iso-resultion
Imposing weak boundary conditions: curved mesh of a propeller (zoom)
Parallel mesh curved subdivision: comparison with standard approaches.