Concave Finite Element Shape Functions

Columbia University, New York NY

Investigators

Abstract

Geometrical concavity in finite elements does not permit the displacement-based shape functions that are restricted to algebraic polynomials. Functions with singularity and rational polynomials (numerator polynomials divided by denominator polynomials), which alleviate the limitation of Taylor polynomials, will be constructed to yield shape functions to guarantee a preassigned degree of continuity within a concave finite element. The application of exact integration (not conventional numerical quadrature schemes) based on the divergence theorem will be employed to integrate the energy density functions leading to element mass and stiffness matrices. In stress-based (hybrid) finite elements, concavity does not pose any additional problems. Theprocedures available for convex polygonal elements will be utilized with appropriate modifications to account for concavity. These novel finite elements are essential in high quality models that necessitate minimum number of tessellations. In addition to domain discretized procedures, three-dimensional boundary element models will utilize the proposed elements to cover boundaries where reentrant corners exist. As an outgrowth of this research, a limited number of curved boundaries, will be pursued to augment the element library. Conventional convex elements with limited forms of geometrical shapes, where isoparametric transformations are routinely employed, will be incorporated in the proposed numerical formulation as special cases. Fundamental mathematical results of projective and perspective geometrical considerations will guide the construction of shape functions including the singular non-classical ones in closed analytical forms. Computer algebra code will symbolically manipulate singular functions as distributions (generalized functions). The weak definition of the square root to generate absolute values, the delta sequences and their formal derivatives to construct Dirac's delta and Heaviside step functions will be seamlessly accommodated within the element shape function and integration routines. All integrals will be interpreted in the distributional sense in a non-classical (weak) formulation. Object-oriented complied codes will result from the computer algebraic constructs based on closed-form manipulations of analytical representations of singularities. This integrated computational environment, which consists of algebraic, numerical and graphics routines, will be employed to generate shape functions and system matrices on both convex and concave domains in order to solve large scale problems where high numerical accuracy cannot be compromised. Application fields will include geotechnical modeling for blasts, soil-structure interactions and bioengineering growth analysis of soft and hard tissues. Second order effects of randomness in constitutive descriptions and boundary geometry will be studied as benchmark examples. High precision applications in the field of high technology, e.g., optical determination of constitutive properties in the micro and lower level material structures, will be explored.