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ME441 Fall 2002 lecture topics |
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| 9/5/02 | 1 | Introductory remarks: FEM discretization of a structural problem, solution, model convergence, advantages, limitations |
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9/9/02
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Matrix analysis: 1-D bar element, assembly of [K], properties of [K], partition of [K], solution procedure |
| 9/11/02 |
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Matrix analysis: beam element, assembly of [K], [K] and [kff] as a function of the boundary cond.s, 2-D bar el. : equilibrium derivation |
| 9/16/02 |
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Matrix analysis: 2-D bar el.: transformation matrix; node numbering for bandwidth min.; node numbering for matrix partition |
| 9/18/02 |
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Matrix analysis: very stiff el.s to approximate reactions and applied displ.s; distributed loads: weight, element expansions, distr. axial load |
| 9/23/02 |
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Matrix analysis: Gaussian elimination and Choleski decomposition for symmetric and banded matrices, storage schemes, use of matrix skyline |
| 9/25/02 |
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Matrix analysis: Gaussian elimination: ill conditioning (geometrical example, truss and beam problems), condition number, scaling |
| 9/30/02 |
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Elasticity: equilibrium equations; continuity conditions and infinitesimal strains; constitutive relations for linear isotropic elastic materials; plane strain and plane stress |
| 10/2/02 |
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Physical interpretation of Gaussian elimination for 1-D problems; Variational principle for conservative systems: definitions, total potential energy, stationarity, examples |
| 10/9/02 |
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Matrix definitions, total potential energy for the elastic continuum, derivation of the principle of virtual works from the equilibrium equations, virtual works and the variational principle, total potential energy at equilibrium. |
| 10/14/02 |
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Rayleigh-Ritz method: assumed displacement field; ex: polynomial solution for bar problem; completeness condition; FE form of R-R; standard FE formulation |
| 10/16/02 |
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Rayleigh-Ritz method: standard FE formulation: bar element, assembly and solution, shape functions for beam element, relationship between nodal d.o.f.'s and shape functions. |
| 10/21/02 |
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Construction of Lagrangian shape functions, 2-D interpolation and nodal shape functions, constant strain triangle (CST); linear 4-noded lagrangian element; element matrices: nodal forces induced by initial stress and strain. |
| 10/23/02 |
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Work equivalent nodal loads due to edge traction
and to concentrated forces; solution satisfaction of equilibrium and compatibility,
inter-element displacement discontinuities due to shape functions mismatch.
PROJECT: input form for nodal and material data. |
| 10/28/02 |
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Project: flow chart, arrays for nodal and elemental
data, equation numbers.
Isoparametric fomulation: 1-D element: mapping from parametric to physical space, strain computation, stiffness matrix. |
| 10/30/02 |
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MIDTERM EXAM |
| 11/4/02 |
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Exam review. Isoparametric formulation: plane element: mapping from parametric to physical space, Jacobian matrix, [B] matrix, stiffness matrix, integration. |
| 11/6/02 |
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Isoparametric formulation: numerical quadrature: Newton-Cotes formula, Gauss quadrature, quadrature in 2-D, order of quadrature for linear and quadratic quadrilateral elements. |
| 11/11/02 |
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Isoparametric formulation: mapping singularities induced by (a) element distortion, (b) midpoint relocation, (c) node sequencing (topology). Isoparametric triangular elements. Stress computation at Gauss points. |
| 11/13/02 |
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Project: stiffness matrix assembly algorithm. Convergence requirements: continuity, constant strain modeling, patch test, rigid body motions, geometrical invariance. |
| 11/18/02 |
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Convergence in general terms; Eigenvalue test for elements: rigid body motions, constant strain and flexural modes; effect of reduced integration: spurious zero-strain modes. Transformations: generalization based on the principle of virtual works. |
| 11/20/02 |
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Transformation matrix: modeling of skewed supports, joining dissimilar elements, multi-point constraints. |
| 11/25/02 |
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Approaches to modeling displacement constraints: transformation matrix, Lagrange multipliers, penalty functions. Example problems. |
| 12/2/02 |
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Symmetry and antisymmetry, symmetrical and antisymmetrical loads, symmetry boundary conditions for displacement and rotations in frames. Symmetry and antisymmetry in continuum domains, boundary conditions, removal of rigid body motions. |
| 12/4/02 |
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Substructuring: domain decomposition, static condensation, partial Gaussian elimination, solution at superelemnt level. Weighted residual approaches: collocation, least squares, Galerkin; unifying concept. |
| 12/9/02 |
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Galerkin weighted residual method: example problem, formulation of uniaxial truss element. Course evaluation. |
| 12/11/02 |
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MIDTERM EXAM #2 |