Why COMSOL use FEM instead of FVM?
Posted July 8, 2021 at 12:49 by Trailokya
Quote:
Originally Posted by mprinkey
FEM lacks a fundamental statement of conservation. FVM (and DG) are axiomatically conservative based on face flux integrals. FEM is defined as a minimization problem--find the solution that best reduces the Galerkin (or Least-Squares) residual of this system. For solid mechanics, that minimization statement makes a lot of sense--configuration of solid mechanical systems map nicely to variational formulations. Conservation equations, however, do not.
For simple flow physics, the difference is not really that important. FEM with linear shape functions *may* be a little more accurate than 2nd order FVM on a per-DOF basis. The FVM code will probably run a bit faster. But, the FVM code will *precisely* (to round-off error) conserve the mass entering and exiting a system. FEM will not be absolutely conservative, without some additional tweaking--using dark arts that I know not. This really becomes an issue with reacting flows, say, where trace concentrations of species can make significant differences. A one-part-in-ten-thousand mass imbalance is inconsequential in external aero or a lid driven cavity, but it could create a dramatic difference in flame shape or attachment points.
Another reason is that FVM solvers are highly optimized for solving flow problems, by basically cutting every corner possible. Segregated solution methods, projection methods, frozen field preconditioning for Newton Krylov...the list is very long. FEM doesn't have these and they do not automatically transfer. FEM tends to do a great job of handling inter-field coupling because it creates a large stiffness matrix using all of the d.o.f.s, solving the system in a coupled manner. And while that is perfect for enforcing solid mechanics constitutive laws, that coupled approach *tends* to be suboptimal from a pure convergence/performance standpoint. The details of these differences are difficult to cover without really getting into the weeds. Suffice to say, FEM methods have grown one way to serve primarily solid mechanics. FVM methods have grown another way (really TWO other ways, as density-based and pressure-based solvers are hugely different in their own right). These decades of accumulated differences has resulted in tool specialization that is hard to overcome.
For simple flow physics, the difference is not really that important. FEM with linear shape functions *may* be a little more accurate than 2nd order FVM on a per-DOF basis. The FVM code will probably run a bit faster. But, the FVM code will *precisely* (to round-off error) conserve the mass entering and exiting a system. FEM will not be absolutely conservative, without some additional tweaking--using dark arts that I know not. This really becomes an issue with reacting flows, say, where trace concentrations of species can make significant differences. A one-part-in-ten-thousand mass imbalance is inconsequential in external aero or a lid driven cavity, but it could create a dramatic difference in flame shape or attachment points.
Another reason is that FVM solvers are highly optimized for solving flow problems, by basically cutting every corner possible. Segregated solution methods, projection methods, frozen field preconditioning for Newton Krylov...the list is very long. FEM doesn't have these and they do not automatically transfer. FEM tends to do a great job of handling inter-field coupling because it creates a large stiffness matrix using all of the d.o.f.s, solving the system in a coupled manner. And while that is perfect for enforcing solid mechanics constitutive laws, that coupled approach *tends* to be suboptimal from a pure convergence/performance standpoint. The details of these differences are difficult to cover without really getting into the weeds. Suffice to say, FEM methods have grown one way to serve primarily solid mechanics. FVM methods have grown another way (really TWO other ways, as density-based and pressure-based solvers are hugely different in their own right). These decades of accumulated differences has resulted in tool specialization that is hard to overcome.
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