Elena Kartasheva

Cellular-functional modeling of heterogeneous objects

The paper presents an approach to modeling heterogeneous objects as multidimensional point sets with multiple attributes (hypervolumes). A theoretical framework is based on a hybrid model of hypervolumes combining a cellular representation and a constructive representation using real-valued functions. This model allows for independent but unifying representation of geometry and attributes, and makes it possible to represent dimensionally non-homogeneous entities and their cellular decompositions. Hypervolume model components such as objects, operations and relations are introduced and outlined. The frameworks inherent multidimensionality allowing, in particular, to deal naturally with time dependence promises to model complex dynamic objects composed of different materials with constructive building of their geometry and attributes. Attributes given at each point can represent properties of arbitrary nature (material, photometric, physical, statistical, etc.). To demonstrate a particular application of the proposed framework, we present an example of multimaterial modeling - a multilayer geological structure with cavities and wells. Another example illustrating the treatment of attributes other than material distributions is concerned with time-dependent adaptive mesh generation where function representation is used to describe object geometry and density of elements in the cellular model of the mesh. The examples have been implemented by using a specialized modeling language and software tools being developed by the authors.

Hybrid Cellular-functional Modeling of Heterogeneous Objects

An approach to modeling heterogeneous objects as multidimensional point sets with multiple attributes (hypervolumes) is presented. Attributes given at each point represent object properties of arbitrary nature (material, physical, etc.). A proposed theoretical framework is based on a hybrid model of geometry and attributes combining a cellular representation and a functionally based constructive representation of dimensionally non-homogeneous entities. Hypervolume model components such as objects, operations and relations are introduced and outlined. We present examples of modeling a multi-layer geological structure with cavities and wells, time-dependent adaptive mesh generation, and conversion of a 3D implicit complex to the cellular representation.

Surface and Volume Discretization of Functionally Based Heterogeneous Objects

The presented approach to discretization of functionally defined heterogeneous objects is oriented towards applications associated with numerical simulation procedures, for example, finite element analysis (FEA). Such applications impose specific constraints upon the resulting surface and volume meshes in terms of their topology and metric characteristics, exactness of the geometry approximation, and conformity with initial attributes. The function representation of the initial object is converted into the resulting cellular representation described by a simplicial complex. We consider in detail all phases of the discretization algorithm from initial surface polygonization to final tetrahedral mesh generation and its adaptation to special FEA needs. The initial object attributes are used at all steps both for controlling geometry and topology of the resulting object and for calculating new attributes for the resulting cellular representation.

Discretization of functionally based heterogeneous objects

The presented approach to discretization of functionally defined heterogeneous objects is oriented towards applications associated with numerical simulation procedures, for example, finite element analysis (FEA). Such applications impose specific constraints upon the resulting surface and volume meshes in terms of their topology and metric characteristics, exactness of the geometry approximation, and conformity with initial attributes. The function representation of the initial object is converted into the resulting cellular representation described by a simplicial complex. We consider in detail all phases of the discretization algorithm from initial surface polygonization to final tetrahedral mesh generation and its adaptation to special FEA needs. The initial object attributes are used at all steps both for controlling geometry and topology of the resulting object and for calculating new attributes for the resulting cellular representation.

Shape recovery using functionally represented constructive models

We propose a method which helps to fit existing parameterized function representation (FRep) models to a given dataset of 3D surface points. Best fitted parameters of the model are obtained by using a hybrid algorithm combining simulated annealing and Levenberg-Marquardt methods. The efficiency of the approach is shown for recovery of two test items. We show through the CAD model processing an application of the proposed approach to the shape recovery followed by finite element mesh generation and adaptation.

Automation of the volumetric models construction

The automation of the function-based (FRep) volumetric modeling task is tackled by introducing template parameterized models and a procedure for recovery of constructive models from segmented point-sets. In order to reuse existing models, we propose to parameterize them and to fit the parameters to different point-sets for optimizing and adapting the shape to different objects of the same class of shapes. The automation of the creation of a constructive FRep model is also considered by creating a recovery procedure for a given segmented pointset and a list of corresponding primitives. A genetic algorithm is used to find the best constructive expression for the object with the given set of primitives in the point cloud segmentation and the set of available operations. The proposed approach is illustrated by fitting of different models to point clouds and by the automatic generation of constructive trees from segmented point-sets for real mechanical parts.

An implicit complexes framework for heterogeneous objects modelling

In this paper we further develop a novel approach for modelling heterogeneous objects containing entities of various dimensions and representations within a cellular-functional framework based on the implicit complex notion. We provide a brief description for implicit complexes and describe their structure including both the geometry and topology of cells of different types. Then the paper focuses on the development of algorithms for set-theoretic operations on heterogeneous objects represented by implicit complexes. We also describe a step-by-step procedure for the construction of a hybrid model using these operations. Finally, we present a case-study showing how to construct a hybrid model integrating both boundary and function representations. Our examples also illustrate modelling with attributes and dynamic modelling.

Heterogeneous objects modelling and rendering using implicit complexes

This paper describes a technology for modelling and rendering heterogeneous objects containing entities of various dimensionalities within a cellular-functional framework based on the implicit complex notion. Implicit complexes make it possible to combine a cellular representation and a constructive function representation. We describe a formal framework for such a hybrid representation and propose a general structure for implicit complexes. Then, we consider how an implicit complex can be described geometrically and topologically along with its associated attributes. Rendering algorithms for implicit complexes using ray-tracing are also discussed. Finally, we present a case study illustrating the proposed methods and algorithms.

Modelling with implicit complexes

This sketch presents a novel method for modelling heterogeneous objects. Such objects can be heterogeneous from the point of view of their dimensionality and their internal structure. We have defined a hybrid model, called an Implicit Complex (IC) [Adzhiev et al. 2001], based on the concepts of cellular spaces and CW-complexes. We have extended CW-complexes with the implicit description of cells and allowed more sophisticated relations, between the cells, to be defined. A heterogeneous object represented by the hybrid model is considered as a single entity integrating different representation schemes with the ability to preserve and extract information about its components and independently process them while guaranteeing topologically correct definitions.