620 research outputs found
The Parma Polyhedra Library: Toward a Complete Set of Numerical Abstractions for the Analysis and Verification of Hardware and Software Systems
Since its inception as a student project in 2001, initially just for the
handling (as the name implies) of convex polyhedra, the Parma Polyhedra Library
has been continuously improved and extended by joining scrupulous research on
the theoretical foundations of (possibly non-convex) numerical abstractions to
a total adherence to the best available practices in software development. Even
though it is still not fully mature and functionally complete, the Parma
Polyhedra Library already offers a combination of functionality, reliability,
usability and performance that is not matched by similar, freely available
libraries. In this paper, we present the main features of the current version
of the library, emphasizing those that distinguish it from other similar
libraries and those that are important for applications in the field of
analysis and verification of hardware and software systems.Comment: 38 pages, 2 figures, 3 listings, 3 table
PARABOLIC BLENDING SURFACES ALONG POLYHEDRON EDGES
In this paper parabolic blending surfaces are defined along a chain of polyhedron edges.
The profile curve of each sweep surface generated for a given edge is a conic section, and
every point of it moves on a conic section around a vertex. According to this, the patches
at the corners are given in rational biquadratic form and they join to the cylindrical
surfaces replacing the edges with 1st order continuity
Certified Roundoff Error Bounds Using Semidefinite Programming.
Roundoff errors cannot be avoided when implementing numerical programs with finite precision. The ability to reason about rounding is especially important if one wants to explore a range of potential representations, for instance for FPGAs or custom hardware implementation. This problem becomes challenging when the program does not employ solely linear operations as non-linearities are inherent to many interesting computational problems in real-world applications. Existing solutions to reasoning are limited in the presence of nonlinear correlations between variables, leading to either imprecise bounds or high analysis time. Furthermore, while it is easy to implement a straightforward method such as interval arithmetic, sophisticated techniques are less straightforward to implement in a formal setting. Thus there is a need for methods which output certificates that can be formally validated inside a proof assistant. We present a framework to provide upper bounds on absolute roundoff errors. This framework is based on optimization techniques employing semidefinite programming and sums of squares certificates, which can be formally checked inside the Coq theorem prover. Our tool covers a wide range of nonlinear programs, including polynomials and transcendental operations as well as conditional statements. We illustrate the efficiency and precision of this tool on non-trivial programs coming from biology, optimization and space control. Our tool produces more precise error bounds for 37 percent of all programs and yields better performance in 73 percent of all programs
Minkowski Sum Construction and other Applications of Arrangements of Geodesic Arcs on the Sphere
We present two exact implementations of efficient output-sensitive algorithms
that compute Minkowski sums of two convex polyhedra in 3D. We do not assume
general position. Namely, we handle degenerate input, and produce exact
results. We provide a tight bound on the exact maximum complexity of Minkowski
sums of polytopes in 3D in terms of the number of facets of the summand
polytopes. The algorithms employ variants of a data structure that represents
arrangements embedded on two-dimensional parametric surfaces in 3D, and they
make use of many operations applied to arrangements in these representations.
We have developed software components that support the arrangement
data-structure variants and the operations applied to them. These software
components are generic, as they can be instantiated with any number type.
However, our algorithms require only (exact) rational arithmetic. These
software components together with exact rational-arithmetic enable a robust,
efficient, and elegant implementation of the Minkowski-sum constructions and
the related applications. These software components are provided through a
package of the Computational Geometry Algorithm Library (CGAL) called
Arrangement_on_surface_2. We also present exact implementations of other
applications that exploit arrangements of arcs of great circles embedded on the
sphere. We use them as basic blocks in an exact implementation of an efficient
algorithm that partitions an assembly of polyhedra in 3D with two hands using
infinite translations. This application distinctly shows the importance of
exact computation, as imprecise computation might result with dismissal of
valid partitioning-motions.Comment: A Ph.D. thesis carried out at the Tel-Aviv university. 134 pages
long. The advisor was Prof. Dan Halperi
Molecular dynamics in arbitrary geometries : parallel evaluation of pair forces
A new algorithm for calculating intermolecular pair forces in molecular dynamics (MD) simulations on a distributed parallel computer is presented. The arbitrary interacting cells algorithm (AICA) is designed to operate on geometrical domains defined by an unstructured, arbitrary polyhedral mesh that has been spatially decomposed into irregular portions for parallelisation. It is intended for nano scale fluid mechanics simulation by MD in complex geometries, and to provide the MD component of a hybrid MD/continuum simulation. The spatial relationship of the cells of the mesh is calculated at the start of the simulation and only the molecules contained in cells that have part of their surface closer than the cut-off radius of the intermolecular pair potential are required to interact. AICA has been implemented in the open source C++ code OpenFOAM, and its accuracy has been indirectly verified against a published MD code. The same system simulated in serial and in parallel on 12 and 32 processors gives the same results. Performance tests show that there is an optimal number of cells in a mesh for maximum speed of calculating intermolecular forces, and that having a large number of empty cells in the mesh does not add a significant computational overhead
Efficient Synthesis of Room Acoustics via Scattering Delay Networks
An acoustic reverberator consisting of a network of delay lines connected via
scattering junctions is proposed. All parameters of the reverberator are
derived from physical properties of the enclosure it simulates. It allows for
simulation of unequal and frequency-dependent wall absorption, as well as
directional sources and microphones. The reverberator renders the first-order
reflections exactly, while making progressively coarser approximations of
higher-order reflections. The rate of energy decay is close to that obtained
with the image method (IM) and consistent with the predictions of Sabine and
Eyring equations. The time evolution of the normalized echo density, which was
previously shown to be correlated with the perceived texture of reverberation,
is also close to that of IM. However, its computational complexity is one to
two orders of magnitude lower, comparable to the computational complexity of a
feedback delay network (FDN), and its memory requirements are negligible
Industrial product design by using two-dimensional material in the context of origamic structure and integrity
Thesis (Master)--Izmir Institute of Technology, Izmir, 2004Includes bibliographical references (leaves: 115)Text in English; Abstract: Turkish and English.xiii, 118 leavesThroughout the history of industrial product design, there have always been attempts to shape everyday objects from a single piece of semi-finished industrial materials such as plywood, sheet metal, plastic sheet and paper-based sheet. One of the ways to form these two-dimensional materials into three-dimensional products is bending following cutting. Similar concepts of this spatial transformation are encountered in the origami form, which has a planar surface in unfolded state, then transforms to a three-dimensional state by folding or by folding following cutting. If so, conceptually it may be useful to think of one-axis bending, which is a manufacturing technique, is somewhat similar to folding paper.In this regard, the studies in the scope of computational origami, which light the way for real-world problems such as how sheets of material will behave under stress, have applications especially in .manufacturing phase. of industrial product design.Besides manufacturing phase, origami design is also used as a product design tool either in .concept creating phase. (in the context of its concepts) or in 'form creating phase' (in the context of its design principles).In this thesis, the designing of industrial products, which are made from sheet material, is presented in a framework that considers the origami design. In the theoretical framework, evolutionary progression of origami design is discussed briefly in order to comprehend the situation of origami design in distinct application fields.Moreover, the elements, principles, basics of origami design and origamic structures are generally introduced. The theoretical framework is completed with the descriptions of the concepts on origami design and origamic structures. In the practical framework, typical applications that have origamic structures in distinct industrial product fields are exemplified. Furthermore, sheet materials and their bending process are taken up separately. By means of its excessive advantages, sheet metal bending is particularly emphasized. The practical framework is completed with several case studies base on sheet metal bending. Finally, the study is concluded with the evaluation of the origamic-structured product in respect of good design principles. Furthermore, designing by considering origami design is recommended to designer to design a good industrial product
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