Geotechnical design and analysis Subject:
Finite Element Method vs Finite Difference Method
Could any one please throw some light on differences between FDM and FEM analysis process.
In my layman term, FDM develops difference equations between grid points in the solution process, so primarily requires rectangular grids. FEM will have shape functions and could have nodes between grid points, so can solve more complex geometry without necessarily requiring rectangular grids. Is this correct ? Are there any other specific differences ?
I think in this way, these are the basics for FEM & FDM principles,
FEM fully discretizes a static problem into a system of algebraic equations with discrete nodal values as the basic unknowns. For the time relay problem, FEM fully discretizes it in spatial domain into ordinary differential equations (ODEs) and solves them with the FD method (Hughes, 1987) The shortcomings of FEM are large demand on computer memory and high computation costs because of the semi-discretization.
In Finite difference methods materials are represented by polyhedral elements within a two/three-dimensional grid that is adjusted by the user to fit the shape of the object to be modeled. Each element behaves according to a prescribed linear or nonlinear stress/strain law in response to applied forces or boundary restraints. The material can yield and flow, and the grid can deform (in large-strain mode) and move with the material that is represented. The explicit, Lagrangian, calculation scheme and the mixed-discretization zoning technique used ensure that plastic collapse and flow are modeled very accurately. Because no matrices are formed, large three-dimensional calculations can be made without excessive memory requirements.
In conclusions, both methods translate a set of differential equations into matrix equations for each element, relating forces at nodes to displacements at nodes. The resulting element matrices, for an elastic material, are identical to those of the finite element method (for constant-strain tetrahedral).
However, they differs in the following respects,
·The “mixed discretization” scheme (Marti and Cundall 1982) is used for accurate modeling of plastic collapse loads and plastic flow. This scheme is believed to be physically more justifiable than the “reduced integration” scheme commonly used with finite elements.
·An “explicit” solution scheme is used (in contrast to the more usual implicit methods). Explicit schemes can follow arbitrary no linearity in stress/strain laws in almost the same computer time as linear laws, whereas implicit solutions can take significantly longer to solve nonlinear problems. Furthermore, it is not necessary to store any matrices, which means: (a) A large number of elements may be modeled with a modest memory requirement; (b) A large-strain simulation is hardly more time-consuming than a small-strain run, because there is no stiffness matrix to be updated.