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The following four-article series was published in a newsletter of the American Society of Mechanical Engineers (ASME). It serves as an introduction to the analysis discipline known as the finite element method (FEM). The author is Steve Roensch, an engineering consultant and expert witness specializing in finite element analysis.
Third in a four-part series.
Previous: Finite Element Analysis (FEA): Pre-processing
The Solution Phase: Solving the Matrix
While the pre-processing and post-processing phases of the finite element method are interactive and time-consuming for the analyst, the solution is often a batch process that is extremely demanding of computer resources.
This phase requires significant processing power. For professional analysis, we recommend using dedicated Mobile Workstations or high-performance Desktop Engineering Computers to handle the large matrix inversions efficiently.
The Governing Equation: Kd = r
The governing equations are assembled into matrix form and are solved numerically. In the case of a linear static structural analysis, the assembled equation is of the form:
[K]{d} = {r}
- K is the system stiffness matrix.
- d is the nodal degree of freedom (dof) displacement vector.
- r is the applied nodal load vector.
To appreciate this equation, one must begin with the underlying elasticity theory. The strain-displacement relation may be introduced into the stress-strain relation to express stress in terms of displacement. Under the assumption of compatibility, the differential equations of equilibrium determine a unique displacement field solution.
Recalling that the expression for the potential energy of an elastic body includes an integral for strain energy stored and integrals for work done by external forces, we can express system potential energy in terms of nodal displacement. Applying the principle of minimum potential energy, we set the partial derivative of potential energy with respect to the nodal dof vector to zero. This results in the summation of element stiffness integrals (K) multiplied by the nodal displacement vector (d) equaling the summation of load integrals (r).
Solver Algorithms
Solution methods for finite element matrix equations are plentiful. Inverting the K matrix directly is computationally expensive and numerically unstable. Instead, specialized algorithms are used:
- Cholesky Factorization: A form of Gauss elimination and a variation on the "LDU" factorization theme. The K matrix is factored into LDU (Lower triangular, Diagonal, Upper triangular). Since L and D are easily inverted, d is determined by back-substitution.
- Wavefront Method: This technique assembles and reduces the equations simultaneously, saving memory.
- Sparse Matrix Techniques: Because node-to-node stiffnesses are non-zero only for nearby node pairs, the stiffness matrix has a large number of zero entries. Modern sparse solvers exploit this to reduce solution time and storage by a factor of 10 or more.
Dynamic Analysis and Advanced Methods
Dynamic analysis often focuses on normal modes. Knowledge of the natural frequencies and mode shapes is sufficient for single-frequency vibration issues. However, when investigating a future product with multiple excited modes, forced response modeling should be used to estimate displacement and dynamic stress at each time step.
This discussion has assumed h-code elements (fixed interpolation polynomials). Other techniques include:
- p-code: Increases the polynomial order iteratively until convergence.
- Boundary Element Method (BEM): Places elements only along the geometrical boundary.
Continue to Part 4:
Finite Element Analysis (FEA): Post-processing
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