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A method for finite element analysis using a regular or structured grid is described that eliminates the need for generating conforming mesh for the geometry. The geometry of the domain is represented using implicit equations, which can be generated from traditional solid models. Solution structures are constructed using implicit equations such that the essential boundary conditions are satisfied exactly. This approach is used to solve boundary value problems arising in thermal and structural analysis. Convergence analysis is performed for several numerical examples and the results are compared with analytical and finite element analysis solutions to show that the method gives solutions that are similar to the finite element method in quality but is often less computationally expensive. Furthermore, by eliminating the need for mesh generation, better integration can be achieved between solid modeling and analysis stages of the design process.