Antonio DiCarlo

Chain-Based Representations for Solid and Physical Modeling

In this paper, we show that the (co)chain complex associated with a decomposition of the computational domain, commonly called a mesh in computational science and engineering, can be represented by a block-bidiagonal matrix that we call the Hasse matrix. Moreover, we show that topology-preserving mesh refinements, produced by the action of (the simplest) Euler operators, can be reduced to multilinear transformations of the Hasse matrix representing the complex. Our main result is a new representation of the (co)chain complex underlying field computations, a representation that provides new insights into the transformations induced by local mesh refinements. Our approach is based on first principles and is general in that it applies to most representational domains that can be characterized as cell complexes, without any restrictions on their type, dimension, codimension, orientability, manifoldness, and connectedness.

Solid and physical modeling with chain complexes

In this paper we show that the (co)chain complex associated with a decomposition of the computational domain, commonly called a mesh in computational science and engineering, can be represented by a block-bidiagonal matrix that we call the Hasse matrix. Moreover, we show that topology-preserving mesh refinements, produced by the action of (the simplest) Euler operators, can be reduced to multi-linear transformations of the Hasse matrix representing the complex. Our main result is a new representation of the (co)chain complex underlying field computations, a representation that provides new insights into the transformations induced by local mesh refinements. This paper is a further contribution towards bridging the subject of computer representations for solid and physical modeling---which flourished border-line between computer graphics, engineering mechanics and computer science with its own methods and data structures---under the general cover of linear algebra and algebraic topology. The main advantage of such an approach is that topology, geometry and physics may coexist in one and the same formalized framework, concurring together to define, represent and simulate the behavior of the model. Our approach is based on first principles and is general in that it applies to most representational domains that can be characterized as cell complexes, without any restrictions on their type, dimension, codimension, orientability, manifoldness, connectedness. Contrary to what might appear at first sight, the theoretical complexity of the present approach is not greater than that of current methods, provided that sparse-matrix techniques with double element access (by rows and by columns) are employed. Last but not least, our tensorbased approach is a significant step forward in achieving close integration of geometrical representations and physics-based simulations, i.e., in the concurrent modeling of shape and behavior.

Technical note: Linear algebraic representation for topological structures

With increased complexity of geometric data, topological models play an increasingly important role beyond boundary representations, assemblies, finite elements, image processing, and other traditional modeling applications. While many graph- and index-based data structures have been proposed, no standard representation has emerged as of now. Furthermore, such representations typically do not deal with representations of mappings and functions and do not scale to support parallel processing, open source, and client-based architectures. We advocate that a proper mathematical model for all topological structures is a (co)chain complex: a sequence of (co)chain spaces and (co)boundary mappings. This in turn implies all topological structures may be represented by a collection of sparse matrices. We propose a Linear Algebraic Representation (LAR) scheme for mod 2 (co)chain complexes using CSR matrices and show that it supports a variety of topological computations using standard matrix algebra, without any overhead in space or running time. A full open source implementation of LAR is available and is being used for a variety of applications.

A codimension-zero approach to discretizing and solving field problems

Computational science and engineering are dominated by field problems. Traditionally, engineering practice involves repeated iterations of shape design (i.e., shaping and modeling of material properties), simulation of the physical field, evaluation of the result, and re-design. In this paper, we propose a specific interpretation of the algebraic-topological formulation of field problems, which is conceptually simple, physically sound, computational effective and comprehensive. In the proposed approach, physical information is attached to an adaptive, full-dimensional decomposition of the domain of interest. Giving preeminence to the cells of highest dimension allows us to generate the geometry and to simulate the physics simultaneously. We will also demonstrate that our formulation removes artificial constraints on the shape of discrete elements and unifies commonly unrelated methods in a single computational framework. This framework, by using an efficient graph-representation of the domain of interest, unifies several geometric and physical finite formulations, and supports local progressive refinement (and coarsening) effected only where and when required.

Fast computation of inertia through affinely extended Euler tensor

Abstract We introduce an affine extension of the Euler tensor which encompasses all of the inertia properties of interest in a convenient linear format, and we show how it transforms under affine maps. This result generalizes the standard theorems on the action of rigid transformations (translations and rotations) on inertia properties, allowing for stretch and shear components of the transformation. More importantly, it provides extremely simple and highly efficient computational tools. By these means, a very fast computation of the inertia properties of polyhedral bodies and surfaces may be obtained. The paper contains some mathematical background, a discussion of the state of the art, and a detailed algorithmic description of the new method, which computes and transforms the Euler tensor (strictly related to the inertia) under general affine maps, through addition and multiplication of 4×4 matrices. Evidence is given that the transformation-based computational technique that is introduced is much faster than conventional domain integration.

PROTO-PLASM: parallel language for adaptive and scalable modelling of biosystems

This paper discusses the design goals and the first developments of Proto-Plasm, a novel computational environment to produce libraries of executable, combinable and customizable computer models of natural and synthetic biosystems, aiming to provide a supporting framework for predictive understanding of structure and behaviour through multiscale geometric modelling and multiphysics simulations. Admittedly, the Proto-Plasm platform is still in its infancy. Its computational framework—language, model library, integrated development environment and parallel engine—intends to provide patient-specific computational modelling and simulation of organs and biosystem, exploiting novel functionalities resulting from the symbolic combination of parametrized models of parts at various scales. Proto-Plasm may define the model equations, but it is currently focused on the symbolic description of model geometry and on the parallel support of simulations. Conversely, CellML and SBML could be viewed as defining the behavioural functions (the model equations) to be used within a Proto-Plasm program. Here we exemplify the basic functionalities of Proto-Plasm, by constructing a schematic heart model. We also discuss multiscale issues with reference to the geometric and physical modelling of neuromuscular junctions.

CAD models from medical images using LAR

ABSTRACTThis paper points out the main design goals of a novel representation scheme of geometric-topological data, named Linear Algebraic Representation (LAR), characterized by a wide domain, encompassing 2D and 3D meshes, manifold and non-manifold geometric and solid models, and high-resolution 3D images. To demonstrate its simplicity and effectiveness for dealing with huge amounts of geometric data, we apply LAR to the extraction of a clean solid model of the hepatic portal vein subsystem from micro-CT scans of a pig liver.

Critical Infrastructures as Complex Systems: A Multi-level Protection Architecture

This paper describes a security platform as a complex system of holonic communities, that are hierarchically organized, but self-reconfigurable when some of them are detached or cannot otherwise operate. Furthermore, every possible subset of holons may work autonomously, while maintaining self-conscience of its own mission, action lines and goals. Each holonic unit, either elementary or composite, retains some capabilities for sensing (perception), transmissive apparatus (communication), computational processes (elaboration), authentication/authorization (information security), support for data exchange (visualization & interaction), actuators (mission), ambient representation (geometric reasoning), knowledge representation (logic reasoning), situation representation and forecasting (simulation), intelligent feedback (command & control). The higher the organizational level of the holonic unit, the more complex and sophisticated each of its characteristic features.

Spatially-Realistic and Reduced Models for Integrative Biomedical Computing

Biomedical computing will greatly benefit from a progressive and adaptive approach to modelling, combined with novel adaptive methods for multiphysics and multiscale simulation. Both symbolic and hierarchical characterizations of the various components should be allowed for, as well as shape reconstruction from high-resolution imaging techniques [2]. Managing finer and finer details and transforming them into additional parameters for coarse-grain models is of the greatest importance. However, it is also essential to be able to analise complicated shapes and patterns, in order to identify their salient features, using computational topology methods based on Morse theory [7]. We apply such ideas to modelling of spatially realistic and reduced domains of structures and ultrastructures of the nervous tissues, where running numerical simulations of the functional behaviour of neurons. In particular, we generate spatially realistic reconstructions of dendrites, axons, glia, and extracellular space domains, using quality surface meshing algorithms to make these reconstructions ready for realistic modeling of dendritic signaling. Reconstruction and modeling tools are used to quantify the variation in surface area and volume of axons, den- drites, glia, extracellular space, synapses, and core subcellular organelles, that could impact electrical signaling. To bridge the gap to onedimensional models that have been used for electrophysiological simulations, we develop appropriately reduced domain models. In this paper we introduce a novel method to compute a minimal fat skeleton, made by hexahedral elements, starting form a point sampling of shape boundary and from the onedimensional and two-dimensional unstable manifolds of the the index 1 and index 2 saddle points of the Morse structure induced by the shape. The result is a cell decomposition with a minimal number of cells, that yet approximate well the shape. The output mesh can be used for simulation of physical behaviour of neural tissue with a minimal number of degrees of freedom.