SUMMARYDomain composition, a recently described method for formulating continuum field problems, removes certain restrictions on the construction of finite element models such that it is possible to solve a finite element problem without using a global compatible mesh. The domain composition method couples or otherwise constrains meshes in local regions to obtain a solution equivalent to that produced by conventional finite element methods. In particular, the domain composition method enables finite element models to be formulated with overlapping elements. Several advantages come from this, including an ability to automatically generate a finite element model from a solid geometric model, an ability to use a variety of element types in a single finite element model and an ability to exactly match element boundaries to the local geometry. This paper shows in detail a finite element formulation of Poisson's equation using domain composition and presents certain key algorithms that incorporate the domain composition method into well-established finite element procedures.
This paper &velops a new method for associating geometxic models, and the operations used to construct them, with the formulation of continuum field equations. Current methods of engineering analysis link a geometric model to the continuum equatioos of a corresponding analysis model via intermediate geometries, or meshes. These meshes "know" little about the global or local geometries.They represent decompositions of the global domain into solvable, non-intemecting subdomains (t%tite elements) that approximate the geometric model. Often the elements of these meshes all share a similar shape, and they tend to impose this shape on the solution of the field equations throughout the model with little regard for the local character of the geometry. Furthermore, it can be a complex and lengthy process to construct a mesh of similarly shaped, non-intersecting elements when the corresponding geometric model has been constructed of intersecting gtmmetric primitives in a variety of shapes.The presented method domain compositi~utilizes the local geometry to parametrize the field equations, capturing the local character of the geometry in the solution to the field equations. Each geometric primitive defms a local coordinate system in which the field equations are formulated. Knowledge of the construction of the global geometric model is used to generate coupling and constraint equations which combme the local formulations into a global fotmuh-ttion.
Presented is a general method for solving sets of nonlinear constraints that include inequalities. Inequality constraints are common in engineering design problems, such as kinematic synthesis. The proposed method uses a modified Newton’s method and introduces a slack variable and a slack constraint to convert each inequality into an equality constraint. Singular value decomposition is used to find the pseudo-inverse of the Jacobian at each iteration. Benefits of this method are that constraint scaling is not an issue and that the method often fails gracefully for inconsistent constraint sets by providing direction for modification of the constraints so that an answer can be found. The method is also competitive with others in terms of the number of function evaluations needed to solve a set of problems taken from the literature.
It is widely recognized that a solid model based on a non-manifold boundary representation can have a more complicated surface topology than one based on a manifold boundary representation, but non-manifold topology has other capabilities that may be more valuable to the application developer. Non-manifold topology can be put to use in existing application areas in ways that differ significantly from the techniques developed for manifold modeling and it can be put to use in new applications that have not been satisfactorily solved by manifold topology. Several applications of non-manifold topology that would be difficult or impossible to implement using a purely manifold geometric modeler are illustrated: automatic formulation of finite element analyses from solid models, automatic generation of machining tool paths for 2½-dimensional pockets, and construction of geometric models using topological constraints. These applications demonstrate how a non-manifold model partitions the entire space in which an object is embedded, preserves elements of the model that would be discarded by conventional schemes, and permits the implementation of a common merge operation. All three applications have been implemented using a two dimensional non-manifold (non-1-manifold) geometric modeler.
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