Adaptive Finite Elements for Systems of PDEs: Software Concepts, Multi-level Techniques and Parallelization

In the recent past, the field of scientific computing has become of more and more importance for scientific as well as for industrial research, playing a comparable role as experiment and theory do. This success of computational methods in scientific and engineering research is next to the enormous improvement of computer hardware to a large extend due to contributions from applied mathematicians, who have developed algorithms which make real life applications feasible. Examples are adaptive methods, high order discretization, fast linear and non-linear solvers and multi-level methods. The application of these methods in a large class of problems demands for suitable and robust tools for a flexible and efficient implementation. In order to play a crucial role in scientific and engineering research, besides efficiency in the numerical solution, also efficiency in problem setup and interpretation of simulation results is of utmost importance. As modeling and computing comes closer together, efficient computational methods need to be applied to new sets of equations. The problems to be addressed by simulation methods become more and more complicated, ranging over different scales, interacting on different dimensions and combining different physics. Such problems need to be implemented in a short period of time, solved on complicated domains and visualized with respect to the demand of the user. %Only a modular abstract simulation environment will fulfill these requirements and allow to setup, solve and visualize real-world problems appropriately. In this work, the concepts and the design of the C++ finite element toolbox AMDiS (adaptive multidimensional simulations) are described. It is shown, how abstract data structures and modern software concepts can help to design user-friendly finite element software, which provides large flexibility in problem definition while on the other hand efficiently solves these problems. Also systems of coupled problems can be solved in an intuitive way. In order to demonstrate its possibilities, AMDiS has been applied to several non-standard problems. The most time-consuming part in most simulations is the solution of linear systems of equations. Multi-level methods use discretization hierarchies to solve these systems in a very efficient way. In AMDiS, such multi-level techniques are implemented in the context of adaptive finite elements. Several numerical results are given which compare this multigrid solver with classical iterative methods. Besides the development of more efficient algorithms also the growing hardware capabilities lead to an improvement of simulation possibilities. Modern computing clusters contain more and more processors and also personal computers today are often equipped with multi-core processors. In this work, a new parallelization approach has been developed which allows the parallelization of sequential code in a very easy way and reduces the communication overhead compared to classical parallelization concepts

Extending STL maps using LBSTs

Associative containers are basic generic classes of the C++ standard library. Access to the elements can be done by key or iterator, but not by rank. This paper presents a new implementation of the map class, which extends the Standard with the ability to support efficient direct access by rank without using extra space. This is achieved using LBST trees. This document reports on the algorithmic engineering of this implementation as well as, experimental results that show its competitive performance compared to the widespread GCC library map implementation.Postprint (published version

Distributed Finite Element Analysis Using a Transputer Network

The principal objective of this research effort was to demonstrate the extraordinarily cost effective acceleration of finite element structural analysis problems using a transputer-based parallel processing network. This objective was accomplished in the form of a commercially viable parallel processing workstation. The workstation is a desktop size, low-maintenance computing unit capable of supercomputer performance yet costs two orders of magnitude less. To achieve the principal research objective, a transputer based structural analysis workstation termed XPFEM was implemented with linear static structural analysis capabilities resembling commercially available NASTRAN. Finite element model files, generated using the on-line preprocessing module or external preprocessing packages, are downloaded to a network of 32 transputers for accelerated solution. The system currently executes at about one third Cray X-MP24 speed but additional acceleration appears likely. For the NASA selected demonstration problem of a Space Shuttle main engine turbine blade model with about 1500 nodes and 4500 independent degrees of freedom, the Cray X-MP24 required 23.9 seconds to obtain a solution while the transputer network, operated from an IBM PC-AT compatible host computer, required 71.7 seconds. Consequently, the $80,000 transputer network demonstrated a cost-performance ratio about 60 times better than the$15,000,000 Cray X-MP24 system

Real-time simulation and visualisation of cloth using edge-based adaptive meshes

Real-time rendering and the animation of realistic virtual environments and characters has progressed at a great pace, following advances in computer graphics hardware in the last decade. The role of cloth simulation is becoming ever more important in the quest to improve the realism of virtual environments. The real-time simulation of cloth and clothing is important for many applications such as virtual reality, crowd simulation, games and software for online clothes shopping. A large number of polygons are necessary to depict the highly exible nature of cloth with wrinkling and frequent changes in its curvature. In combination with the physical calculations which model the deformations, the effort required to simulate cloth in detail is very computationally expensive resulting in much diffculty for its realistic simulation at interactive frame rates. Real-time cloth simulations can lack quality and realism compared to their offline counterparts, since coarse meshes must often be employed for performance reasons. The focus of this thesis is to develop techniques to allow the real-time simulation of realistic cloth and clothing. Adaptive meshes have previously been developed to act as a bridge between low and high polygon meshes, aiming to adaptively exploit variations in the shape of the cloth. The mesh complexity is dynamically increased or refined to balance quality against computational cost during a simulation. A limitation of many approaches is they do not often consider the decimation or coarsening of previously refined areas, or otherwise are not fast enough for real-time applications. A novel edge-based adaptive mesh is developed for the fast incremental refinement and coarsening of a triangular mesh. A mass-spring network is integrated into the mesh permitting the real-time adaptive simulation of cloth, and techniques are developed for the simulation of clothing on an animated character

A Well-Designed, Tree-Based, Generic Map Component to Challenge the Progress towards Automated Verification

This thesis presents a non-trivial candidate software component assembly that presents an opportunity and a challenge to the progress towards automated verification. It presents an opportunity because the data abstraction implementation can serve as a proof of concept of the idea that well-designed and well-annotated software components with mathematical specifications and well-engineered implementation(s) lead to generated verification conditions (VCs) of correctness that are obvious to prove. It presents a challenge because verification of the implementation involves multiple theories and the use of a tree concept that is based on a general tree theory for which there are no special-purpose solvers. The thesis contains a specification for a conceptualization of a tree with a position that makes it easy to explore and navigate a tree even as it avoids any explicit references to simplify reasoning. The thesis also contains concept enhancements for trees and an implementation layered using trees for a data abstraction for searching (a version of maps). A key contribution is the development of the implementation so that it is amenable for verification with internal assertions such as representation invariants and abstraction relations, operation specifications, loop invariants, and progress metrics, all of which involve the general tree theory

Robot graphic simulation testbed

The objective of this research was twofold. First, the basic capabilities of ROBOSIM (graphical simulation system) were improved and extended by taking advantage of advanced graphic workstation technology and artificial intelligence programming techniques. Second, the scope of the graphic simulation testbed was extended to include general problems of Space Station automation. Hardware support for 3-D graphics and high processing performance make high resolution solid modeling, collision detection, and simulation of structural dynamics computationally feasible. The Space Station is a complex system with many interacting subsystems. Design and testing of automation concepts demand modeling of the affected processes, their interactions, and that of the proposed control systems. The automation testbed was designed to facilitate studies in Space Station automation concepts

Sparse Volumetric Deformation

Volume rendering is becoming increasingly popular as applications require realistic solid shape representations with seamless texture mapping and accurate filtering. However rendering sparse volumetric data is difficult because of the limited memory and processing capabilities of current hardware. To address these limitations, the volumetric information can be stored at progressive resolutions in the hierarchical branches of a tree structure, and sampled according to the region of interest. This means that only a partial region of the full dataset is processed, and therefore massive volumetric scenes can be rendered efficiently. The problem with this approach is that it currently only supports static scenes. This is because it is difficult to accurately deform massive amounts of volume elements and reconstruct the scene hierarchy in real-time. Another problem is that deformation operations distort the shape where more than one volume element tries to occupy the same location, and similarly gaps occur where deformation stretches the elements further than one discrete location. It is also challenging to efficiently support sophisticated deformations at hierarchical resolutions, such as character skinning or physically based animation. These types of deformation are expensive and require a control structure (for example a cage or skeleton) that maps to a set of features to accelerate the deformation process. The problems with this technique are that the varying volume hierarchy reflects different feature sizes, and manipulating the features at the original resolution is too expensive; therefore the control structure must also hierarchically capture features according to the varying volumetric resolution. This thesis investigates the area of deforming and rendering massive amounts of dynamic volumetric content. The proposed approach efficiently deforms hierarchical volume elements without introducing artifacts and supports both ray casting and rasterization renderers. This enables light transport to be modeled both accurately and efficiently with applications in the fields of real-time rendering and computer animation. Sophisticated volumetric deformation, including character animation, is also supported in real-time. This is achieved by automatically generating a control skeleton which is mapped to the varying feature resolution of the volume hierarchy. The output deformations are demonstrated in massive dynamic volumetric scenes

Workshop on the Integration of Finite Element Modeling with Geometric Modeling

The workshop on the Integration of Finite Element Modeling with Geometric Modeling was held on 12 May 1987. It was held to discuss the geometric modeling requirements of the finite element modeling process and to better understand the technical aspects of the integration of these two areas. The 11 papers are presented except for one for which only the abstract is given