HiGeoM: A symbolic framework for a unified function space representation of trivariate solids for isogeometric analysis

• We propose a unified function space representation of trivariate solids and its attributes.
• We develop an approach for ab-initio construction of trivariate description of geometry using generative modeling techniques.
• The analysis of the constructed geometries enables isogeometric analysis without need for finite element mesh.
• We develop a symbolic modeling framework to construct trivariate solids and carry out isogeometric analysis.

In this paper, a procedural description of solids is proposed with representational domain that encompasses the needs in Computer Aided Design (CAD), heterogeneous object modeling and Computer Aided Engineering (CAE). Specifically, a function space formalism, that unifies the representation of geometry and physical attributes, is proposed along with an algebra that operates on elements in those spaces. A declarative programming language is provided, which accepts a subset of mathematical semantics specified using the Mathematical Markup Language (MathML) as well as a fortran-like syntax. The language enables a high-level definition of the geometric model using continuous affine transformation and boolean operators leading to a mixed-dimensional representation. Thus, it is possible to construct analytically defined shapes as well as spline representation of complex geometry. The framework is demonstrated by constructing complex engineered solids and isogeometrically solving boundary value problems on the constructed solid.