Increasing complexity of both geometrical and physical domains results in near-intractable computational times for realistic simulations. It is thus necessary to reduce the size of the discretized problems. Mesh adaptation has become nowadays a powerful tool to improve the discrete representation of complex solution fields in many applications. Adapting the mesh may lead to computational overheads, as well as complex algorithmic and software developments. This motivates the quest for efficient and accessible methods. Model Order Reduction (MOR) is another way to reduce the size of the discretized problem for parametric problems by considering smaller solution spaces.
r-adaptation for embedded geometries and fronts
r-adaptation techniques are an appealing approach for unsteady simulations of sharp moving fronts. Compared to h-refinement, one of their attractive characteristics is the relative algorithmic simplicity, and the ease of defining conservative remaps due to the inherent continuous nature of the process.
The method was beneficial in the modelling of the 2011 Tohoku tsunami, and was then extended to global simulations on the sphere.
Barotropic instability. Relative vorticity field with corresponding adapted meshes at different times.
Anisotropic adaptation (h-adaptation)
Embedded geometries
Unfitted discretizations are becoming quite widespread for the geometrical flexibility they offer. The accuracy with which boundary conditions are met is directly linked to the accuracy of the implicit description of the solid, which is typically done by means of a level-set function. In order to control the error in the geometric representation and in the numerical solution, mesh adaptation can be performed with respect to a metric related to the level-set function and to the error estimator of the numerical solution.
Laminar flow over a delta wing. Slices of the Mach number and adapted mesh.
Laminar flow over a delta wing. lComparison flow – mesh at the trailing edge.
Parallel remeshing
Computational mechanics solvers nowadays routinely exploit parallel, distributed memory computer architectures, raising the need for generating and adapting larger and larger meshes. Sequential remeshing in a parallel simulation represents a significant performance bottleneck. Parallel remeshing is thus becoming increasingly demanded in large scale simulations.
The ParMmg library is built on top of the Mmg3d remesher to provide parallel
tetrahedral mesh adaptation in a free and open source software. Among the many possible remeshing parallelization methods, a modular approach is adopted by selecting an iterative remeshing-repartitioning scheme that does not modify the adaptation kernel.
The sequential remeshing kernel is applied at each iteration on the interior partition of each process while maintaining fixed (non-adapted) parallel interfaces. Then the adapted mesh is repartitioned in order to move the non-adapted frontiers to the interior of the partitions at the next iteration.
Adaptation to a complex analytical pattern, on 1 and 1024 CPU cores.
Accessible implementations
Anisotropic mesh adaptation has proved since the early 1990s its ability to reduce the computational cost of numerical simulations while increasing their accuracy in many contexts. Yet, it remains reserved to a small circle of users. One of the main reasons for this it that it remains complicated to set up efficient adaptive frameworks. A full adaptive pipeline is made of several specialized blocks, that require specific knowledge to develop or even use. There are few open-source software bricks available, and they are often difficult to use or incompatible with other software choices.
We have been committed to providing ready-to-use software solution to the general scientific public. Mmg includes various open-source tools to facilitate mesh adaptation. An ongoing effort is made to include Mmg in the widely used scientific library PETSc, and develop a full adaptive framework within the library.
Model Order Reduction
A wealth of applications in science and engineering involve the solution to computational fluid dynamics (CFD) problems for many different system configurations. For this class of problems, it is important to reduce the marginal cost associated with a given simulation over a range of parameters. Model order reduction (MOR) techniques rely on an offline/online decomposition to reduce marginal costs. During the offline phase, we rely on high-fidelity (hf) simulations to generate a reduced-order model (ROM) to estimate the solution over a range of parameters. During the online or deployment phase, given a new value of the parameter, we query the ROM to estimate the solution field.
MOR techniques rely on the projection of the equation onto a unique shared hf discretisation and thus rely on the assumption that the underlying hf discretisation is accurate for all parameters in a prescribed range: for problems with parameter-dependent shocks, this requires accurate adaptive mesh refinement over a broad portion of the spatial domain and is often unfeasible.
We expect that the introduction of anisotropic mesh adaptation techniques in MOR workflows will result in considerable computational gain in the offline stage, and contribute to lifting current locks. This is the topic of the Inria Exploratory Action AM²OR.
