SUMMARYA technical description of the algorithms employed in the modified quadtree mesh generator is .given. Although the basis of the mesh generator is the same as the original version developed by Yerry and Shephard,'.2 the actual algorithms on which it is built have been entirely changed for the purpose of ensuring the robustness of the technique. As demonstrated in the paper the algorithmic changes made do ensure the robustness of the approach, but introduce additional algorithmic difficulties, the solutions of which are also presented. In addition to examples showing the capability of the mesh generator, the linear computational growth rate of the mesh generator is demonstrated.

Abstract. This paper discusses an automatic, adaptive finite element modeling system consisting of mesh generation, finite element analysis, and error estimation. The individual components interact with one another and efficiently reduce the finite element error to within an acceptable value and perform only a minimum number of finite element analyses.One of the necessary.components in the automated system is a multiple-level local remeshing algorithm. Given h-refinement information provided by an a posteriori error estimator, and adjacency information available in the mesh data structures, the local remeshing algorithm grades the refinement toward areas requesting refinement. It is shown that the optimal asymptotic convergence rate is achieved, demonstrating the effectiveness of the intelligent multiple-level local h-refinement.

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SUMMARYHydrated soft tissues of the human musculoskeletal system, such as articular cartilage in diarthrodial joints or the annulus fibrosis of the intervertebral disk, are accurately represented by a biphasic continuum model consisting of an incompressible solid phase (collagen and proteoglycan) and incompressible, inviscid fluid (interstitial fluid) and derive from the continuum theory of mixtures. These tissues exhibit a viscoelastic-type response which is caused by the diffusive drag of the fluid phase as it flows past the solid phase. In this study an automated, adaptive, finite element solution of the governing biphasic equations is presented. The finite element formulation is based on a mixed-penalty approach in which the penalty form of the continuity equation for the mixture is included in the weak form. Pressure, solid and fluid velocities are interpolated independently, and the coefficients of the C -' pressure field are eliminated at the element level. The resulting matrix form is a system of first order differential equations which is solved via standard finite difference methods. Mesh generation and updating, including both refinement and coarsening, for the two-dimensional examples examined in this study are performed using Finite Quadtree. The adaptive analysis is based on an error indicator which is the L2 norm of the difference between the finite element solution and a projected finite element solution. Total stress, calculated as the sum of the solid and fluid phase stresses, is used in the error indicator. Rezoning is accomplished by transferring the finite element solution for the primary variables onto the locally updated mesh using a projected field. These projected values allow the finite difference algorithm to proceed in time using the updated mesh. The accuracy and effectiveness of this adaptive finite element analysis is demonstrated using two-dimensional axisymmetric problems corresponding to the unconfined compression of a cylindrical disk of soft tissue and the indentation of a thin sheet of soft tissue. The method is shown to effectively capture the steep gradients in both problems and to produce solutions in good agreement with independent, converged, numerical solutions.

SUMMARYThe ability of the finite octree and finite quadtree automatic mesh generators to satisfy effectively a wide variety of meshing capabilities is demonstrated. Areas considered include the ability to integrate easily with geometric modelling systems, versatile interactive mesh control functions, the ability to reorder unknowns directly from the finite element data structure and the ability to support automated finite element modelling.