UBC Theses and Dissertations
Convergence of mixed methods in continuum mechanics and finite element analysis Mirza, Farooque Aguil
The energy convergence of mixed methods of approximate analysis for problems involving linear self-adjoint operators is investigated. A new energy product and the associated energy norm are defined for such indefinite systems and then used in establishing the strain energy convergence and estimation of error for problems in continuum mechanics. In the process, the completeness requirements are laid out for approximate solutions. Also established is the mean convergence of the basic variable e.g. displacements and stresses. After accomplishing a new mathematical framework for the mixed methods in continuum, the theory is then extended to the finite element method. The completeness requirements, convergence criteria and the effect of continuity requirements on convergence are established. The flexibility offered by the mixed methods in incorporating the boundary con ditions is also demonstrated. For stress singular problems, the strain energy convergence is established and an energy release method for determining the crack intensity factor K. is presented. A detailed eigenvalue-eigenvector analysis of the mixed finite element matrix is carried out for various combinations of interpolations for the plane stress linear elasticity and the linear part of the Navier-Stokes equations. Also discussed is its relation to the completeness requirements. Finally, numerical results are obtained from applying the mixed finite element method to several examples. These include beam bending, a plane stress square plate with parabolically varying end loads, a plane stress cantilever and plane strain stress concentration around a circular hole. A plane stress example of a square plate with symmetric edge cracks is also solved to study the strain energy convergence. Lastly, two rectangular plates, one with symmetric edge cracks and the other with a central crack are considered to determine the crack intensity factor K. In most of the examples, the strain energy convergence rates are predicted and compared with the numerical results, and excellent agreement is observed.
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