CutFEM: seeing the whole with the sum of the parts
Geometry can describe very complex shapes, such as fluted curves or metallic mesh. Chemistry and physics can describe properties such as temperatures, velocities, and concentrations or flexibility and elasticity. How do you combine these seemingly separate ways of seeing objects? And what if those objects change over time? A successful method would allow researchers to predict material failures, simulate fluids in changing domains, and design products with optimized properties.
It’s often said that to solve a problem (mathematical or otherwise), it’s easiest to break it down into smaller pieces. Then it’s possible to address each piece, one at a time.
That’s part of the strategy for the project CutFEM: A team of researchers based at Umeå University intends to use numerical methods to “cut” apart large and complex geometric models into smaller pieces with simple geometry. Models of physical phenomena can be formulated using these smaller pieces, which then can be folded together again to see the whole picture.
The Umeå team is working to use this modeling method to look at a variety of materials and objects, while improving the formulations in CutFEM to make it more accurate and useable.
When modeling new materials and shapes, researchers want to see not only how an object’s outer surface changes, say, with regard to its shape, but how that might affect its other characteristics — mass or stiffness, thickness or stretchiness — and vice versa. That whole picture might be very complex. Examples include calculating the interactions between two objects that contain fluid within thin membranes, or the internal structure of a metal antenna with branching arms.
The Umeå team would like to be able to look at an elastic structure and find its optimal stiffness, given a certain amount of material. While commercial software can solve this problem, it typically generates unrealistic preliminary designs — the result could be unseen weak spots or fatigue at inward corners or stress points. A more precise mathematical formulation together with a more detailed description of the geometry of the object could lead to the accurate description of these boundaries.
Using this piecemeal method, CutFEM could also provide better representations of objects where the best design is very thin; these are difficult to represent using typical methods, draping the thin material on a background mesh using a density function.
“We have developed a new CutFEM formulation for thin materials exposed to acoustic fields. As opposed to the standard formulation, this method can seamlessly handle the situation when the material distribution concentrates on a surface, a property that will be essential for shape optimization, in particular, in acoustic applications,” says Martin Berggren, professor of computing science at Umeå University.
Researchers could use these modeling results to create acoustic devices, for example, a speaker that takes up as little room as possible in a car. Anything that receives waves — sound, for example, or radio signals to antenna — could be designed to use as little material as efficiently as possible, using complex geometrical patterns that can be accurately described with partial differential equations, cut apart into digestible chunks.
“CutFEM provides a solution to effective and robust discretization of complex geometries, which is the major bottleneck in industrial simulation, accounting for more than 80% of the time spent on finite element simulation in industry. Thus all applications where the geometry of the computational domain is complicated are suitable for CutFEM. An important example is stress analysis of complicated mechanical components and design optimization of the geometry. Another area of interest is fluid structure interaction with large deformations,” such as the valves in the human heart, says Mats G. Larson, principal investigator for CutFEM and professor of mathematics at Umeå University.