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Best Curved Adapted Meshes for Space-Time Flow Simulations

Periodic Reporting for period 4 - Tesseract (Best Curved Adapted Meshes for Space-Time Flow Simulations)

Reporting period: 2021-10-01 to 2022-09-30

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.
The Main Objective of the TESSERACT project is to generate optimal quality curved adapted meshes for space-time flow simulations with unstructured high-order methods. To this end, we have reached the following achievements concerning the project objectives.

(O1) Curved meshing for adaptivity:

- Automatically imposing incremental boundary displacements for valid mesh morphing and curving.

- We present a nodal interpolation method to approximate a subdivision model.

- We answer the questions of the high-order technology focus group (HO-TFG) about the mesh generation for the high-lift common research model of the 4-th high-lift prediction workshop.

(O2) Anisotropic resolution for curved meshes:

- Defining a regularized shape distortion (quality) measure for curved high-order 2D elements on a Riemannian plane.

- An optimization method to adapt straight-edged and curved piece-wise polynomial meshes to the stretching and alignment of a target metric.

- We detail how to use Newton’s method for distortion-based curved r-adaption to a discrete high-order metric field.

(O3) Quality measures for curved meshes:

- Imposing boundary conditions to match a CAD virtual geometry for the mesh curving problem.

- We present a distributed parallel mesh curving method for virtual geometry.

- We present a specific-purpose solver to approximate curves with super-convergent rates.

- We present a new disparity functional to measure and improve the geometric accuracy of a curved high-order mesh that approximates a target geometry model.

(O4) Adapted 4D curved meshing for space-time flow simulation:

- To refine meshes, we have proposed a parallel distributed approach that preserves the curvature of a target geometry.

- Local bisection for refinement of 4D simplicial meshes. We have proposed a recursive refine to conformity procedure in two stages.

- A simple tool to visualize 4D unstructured pentatopic meshes. The results show that the method is suitable to explore 4D unstructured meshes visually.

- To locally refine, we propose a new method to mark for bisection the edges of an arbitrary 3D unstructured conformal mesh.

- We present an n-dimensional marked bisection method for unstructured conformal meshes. We devise the method for local refinement in adaptive n-dimensional applications.
All the results achieved so far have been peer-reviewed and therefore, the following list represents our progress beyond the state of the art:

-An augmented Lagrangian formulation to impose boundary conditions for distortion based mesh moving and curving
-Subdividing triangular and quadrilateral meshes in parallel to approximate curved geometries
-Automatically imposing incremental boundary displacements for valid mesh morphing and curving
-Local Bisection for Conformal Refinement of Unstructured 4D Simplicial Meshes
-Defining a Stretching and Alignment Aware Quality Measure for Linear and Curved 2D Meshes
-Imposing Boundary Conditions to Match a CAD Virtual Geometry for the Mesh Curving Problem
-Isometric Embedding of Curvilinear Meshes Defined on Riemannian Metric Spaces
-Pre-conditioning and continuation for parallel distributed mesh curving
-Measuring and improving the geometric accuracy of piece-wise polynomial boundary meshes
-Generation of Curved Meshes for the High-Lift Common Research Model
-An Efficient Solver to Approximate CAD Curves with Super-Convergent Rates
-High-Order Metric Interpolation for Curved R-Adaption by Distortion Minimization
-Bisecting with Optimal Similarity Bound on 3D Unstructured Conformal Meshes
-Automatic Penalty and Degree Continuation for Parallel Pre-Conditioned Mesh Curving on Virtual Geometry
-Conformal Marked Bisection for Local Refinement of n-Dimensional Unstructured Simplicial Meshes
-Interpolation of Subdivision Features for Curved Geometry Modeling
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.