Meshing and Solving the Finite Element Equation
Main First Topic: Matrix Operations – Solution of the Finite Element Equation Ku = F
Gauss elimination = Triangularization + Back-substitution
Iterative Solver + Optimization (Jacobian Conjugate Gradient JCG Method)
Lanczos (Eigenvalue Problems – Dynamics)
Power Dynamics Extractor
Main Second Topic: Element Selection
Lagrange polynomias order 1,2,3 elements (Zienkiewicz p. 267)
Element selection and cantilever beams
The 1D basic element k = (AE/L) [[1 -1], [-1 1]] and its use in building truss models (need for
coordinate transformation to create the stiffness matrix of inclided elements – From local to global
stiffness)
2D triangle elements orders 1 and 2
2D quadrilateral elements orders 1 and 2
Example: Case of the buckling ladder in NYC
Element Technology – contd.
3D tetrahedral elements orders 1 and 2
3D brick elements orders 1 and 2
Four-nodes shell elements orders 1 and 2
Example: Clamped Al (E=10e6, nu=0.3) Square Plate 10 x 10 x 0.1 in under 100 psi normal pressure ->
displ = 1.376 in; stress = 307.8 ksi Use of shell elements, orders 1, 2
Use of solid elements, tetrahedral order, 2, brick, order 2
Meshing Guidelines
Use quadratic elements whenever possible (Linear Case)
Use linear elements quads/bricks (non-linear problems)
Mapped Mesh vs Free Mesh (2D)
Concatenation to create mapped mesh into an L shape
3D – 6 sides figures easy to mesh w/mapped mesh
Concatenation of Areas to create mapped meshes OR
Sweeping (preferred when there is a front and a back with same topology) - Steps
1) Mesh complex face with shells
2) Sweep mash over the volume
3) Clear shells
Complex Model
Import CAD files (IGES – poor, .SAT – OK, .STEP (only though workbench -> .ANF)
Example: Al pipe (20 in internal radius, 2 in tall) attached to a flange (2.5 in long) on one end, both 0.25
in thick; flange has 20 holes w/bolts. The non-flange end of the pipe is constrained, pressure pulls the
flange on the other end.
1/40 sector model w/rotating nodal displacement system (including a filled at pipe-flange joint)