Periodic Reporting for period 1 - X-MESH (An eXtreme Mesh deformation method to follow sharp physical interfaces)
Reporting period: 2023-09-01 to 2025-02-28
This project will develop an innovative approach, X-MESH, to overcome a major difficulty associated with engineering analysis: we aim to provide a revolutionary way to track physical interfaces in finite element simulations using extreme deformation of the meshes. Unprecedented low computational cost, high robustness and accuracy are expected as the proposed approach is designed to avoid the pitfalls of the current methods, especially for topological changes.
The key idea of the project has emerged in a synergistic brainstorming between J.-F. Remacle (meshing methods) and N. Moe ̈s (computational methods). This key idea is to allow elements to deform up to zero measure. For example, a triangle can deform to an edge or even a point. This idea is rather extreme and totally revisits the interaction between the meshing community and the computational community who, for decades, have striven to interact through beautiful meshes.
Six areas in fluid and solid mechanics as well as heat transfer are targeted. Interfaces will be either (i) material, i.e. attached to particles of matter (the interface between two immiscible fluids or the dry interface in a wetting and drying model) (ii) immaterial, i.e. migrating through the material (a solidification front, contact front, yield front in yield stress fluid flow or a crack front).
Successes brought by X-MESH are expected in the following engineering areas: safety design and maintenance, manufacturing processes, coastal engineering, energy efficiency, ocean modeling to cite a few. The project takes place in a stimulating environment mixing senior staff with PhDs and Post-docs to produce and disseminate publications with open source pieces of software. It also intends to create a synergy at large with the computational communities dealing with interfaces and fronts in all fields of science: topology optimization, superconductivity, tissue growth, hydrogel swelling, crystal growth, ferroelectric crystal...
The key idea of the project has emerged in a synergistic brainstorming between J.-F. Remacle (meshing methods) and N. Moe ̈s (computational methods). This key idea is to allow elements to deform up to zero measure. For example, a triangle can deform to an edge or even a point. This idea is rather extreme and totally revisits the interaction between the meshing community and the computational community who, for decades, have striven to interact through beautiful meshes.
Six areas in fluid and solid mechanics as well as heat transfer are targeted. Interfaces will be either (i) material, i.e. attached to particles of matter (the interface between two immiscible fluids or the dry interface in a wetting and drying model) (ii) immaterial, i.e. migrating through the material (a solidification front, contact front, yield front in yield stress fluid flow or a crack front).
Successes brought by X-MESH are expected in the following engineering areas: safety design and maintenance, manufacturing processes, coastal engineering, energy efficiency, ocean modeling to cite a few. The project takes place in a stimulating environment mixing senior staff with PhDs and Post-docs to produce and disseminate publications with open source pieces of software. It also intends to create a synergy at large with the computational communities dealing with interfaces and fronts in all fields of science: topology optimization, superconductivity, tissue growth, hydrogel swelling, crystal growth, ferroelectric crystal...
In this first period, we developed the X-MESH method in a number of areas
1) A general framework for handling zero-size finite elements. We call this method TFEM for tempered finite elements.
2) The application of X-MESH to various relevant engineering problems: i) the Stefan problem (phase change)
ii) nonlinear diffusion, otherwise known as the porous media equation iii) contact problems in elasticity, iv)
shock problems in fluid mechanics, v) threshold fluids, vi) two-phase fluids and vii) crack propagation problems.
3) The development of meshing techniques for front tracking within the X-MESH framework.
4) The development of X-MESH in space-time, including the generation of meshes in dimension 3 (x,y,t) and 4 (x,y,z,t).
1) A general framework for handling zero-size finite elements. We call this method TFEM for tempered finite elements.
2) The application of X-MESH to various relevant engineering problems: i) the Stefan problem (phase change)
ii) nonlinear diffusion, otherwise known as the porous media equation iii) contact problems in elasticity, iv)
shock problems in fluid mechanics, v) threshold fluids, vi) two-phase fluids and vii) crack propagation problems.
3) The development of meshing techniques for front tracking within the X-MESH framework.
4) The development of X-MESH in space-time, including the generation of meshes in dimension 3 (x,y,t) and 4 (x,y,z,t).
A significant number of research results beyond the state of the art were produced in this first period.
1) A rigorous theoretical framework to extend the set of admissible meshes for the finite element method.
This method, called TFEM for tempered finite elements, has already had concrete applications in the fields
mentioned in the previous point, but will have a large number of extensions -- contact, discontinuous finite
elements without edge terms, negative volume elements...
2) The development of innovative variational formulations which not only involve the fields (classical energy using temperature,
displacement, etc. as variables) but also the geometry of the domain (configurational energy depending on the position of the mesh
nodes and its topology). A remarkable result of these new formulations is that the minimum of this energy corresponds to strict conservation
of quantities that are difficult to calculate, such as stress intensity factors in fracture mechanics.
3) Space-time methods coupled with X-MESH are clearly one of our main focuses. We have demonstrated that it is possible to obtain space-time
mesh adaptation that perfectly maintains field continuity when the mesh is modified (geometry and topology). These methods are now being applied
to “high-speed” fluid mechanics with shocks, as well as to other physics problems of interest.
We have already been contacted by European industries (Safran in the first instance) who are already following our developments.
1) A rigorous theoretical framework to extend the set of admissible meshes for the finite element method.
This method, called TFEM for tempered finite elements, has already had concrete applications in the fields
mentioned in the previous point, but will have a large number of extensions -- contact, discontinuous finite
elements without edge terms, negative volume elements...
2) The development of innovative variational formulations which not only involve the fields (classical energy using temperature,
displacement, etc. as variables) but also the geometry of the domain (configurational energy depending on the position of the mesh
nodes and its topology). A remarkable result of these new formulations is that the minimum of this energy corresponds to strict conservation
of quantities that are difficult to calculate, such as stress intensity factors in fracture mechanics.
3) Space-time methods coupled with X-MESH are clearly one of our main focuses. We have demonstrated that it is possible to obtain space-time
mesh adaptation that perfectly maintains field continuity when the mesh is modified (geometry and topology). These methods are now being applied
to “high-speed” fluid mechanics with shocks, as well as to other physics problems of interest.
We have already been contacted by European industries (Safran in the first instance) who are already following our developments.