Shape optimization of elasto-plastic bodies under plane strains: Sensitivity analysis and numerical implementation

Optimal shape design problems for an elastic body made from physically nonlinear material are presented. Sensitivity analysis is done by differentiating the discrete equations of equilibrium. Numerical examples are included.Notation U ^ad set of admissible continuous design parameters - U _h ^ad set of admissible discrete design parameters - function fromU _h ^ad defining shape of body - _h function fromU _h ^ad defining approximated shape of body - vector of nodal values of _h - { _n} sequence of functions tending to - () domain defined by - K bulk modulus - shear modulus - penalty parameter for contact condition - V() space of virtual displacements in() - V _h(_h) finite element approximation ofV() - J cost functional - J _h discretized cost functional - J algebraic form ofJ _h - (u) stress tensor - e(u) strain tensor - K stiffness matrix - f force vector - b(q) term arising from nonlinear boundary conditions - q vector of nodal degrees of freedom - p vector of adjoint state variables - J Jacobian of isoparametric mapping - |J| determinant ofJ - N vector of shape function values on parent element - L matrix of shape function derivatives on parent element - G matrix of Cartesian derivatives of shape functions - X matrix of nodal coordinates of element - D matrix of elastic coefficients - B strain-displacement matrix - _P part of boundary where tractions are prescribed - _u part of boundary where displacements are prescribed - variable part of boundary - strain invariant     

Speicheroptimale Formelübersetzung

Summary We investigate the problem of translating expressions optimally, i.e. such that at running time they need as few memory cells as possible and that they can be evaluated fast. An efficient (recursive) algorithm is presented (there is always one based on trial and error-methods) for the case of independent inputs and operators to an expression. This includes the case that the result of an operation needs not only one but an arbitrary nonnegative number of auxiliary memory cells which may be determined at compile-time.In [5, 7, 8 and 1] it is shown that it is always possible to compose an optimal translation of an expression ₁ o ₂ by arbitrary but optimal translations of ₁ and ₂ if this number is constant. We prove now the existence of special optimal translations of ₁ and ₂ which always can be composed to an optimal translation of ₁ o ₂ in the general case. This is the hinge for a recursive algorithm.

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Abschließend sei es mir erlaubt, Herrn Prof. Dr. H. Walter für die Anregung zu dieser Arbeit und Frau Dr. B. Schinzel für die hilfreichen Gespräche in diesem Zusammenhang herzlich zu danken.     

Efficient Solution of Multi-Term Fractional Differential Equations Using P(EC)<Superscript><Emphasis Type="Italic">m</Emphasis></Superscript>E Methods

We investigate strategies for the numerical solution of the initial value problem with initial conditions where 0<₁<₂<<. Here y ⁽ _j ) denotes the derivative of order _j >0 (not necessarily _j ) in the sense of Caputo. The methods are based on numerical integration techniques applied to an equivalent nonlinear and weakly singular Volterra integral equation. The classical approach leads to an algorithm with very high arithmetic complexity. Therefore we derive an alternative that leads to lower complexity without sacrificing too much precision. Mathematics Subject Classification: Primary 65L05; Secondary 65L06, 65L20     

The co-invariant generator: An aid in deriving loop bodies

Given a loop invariant,I, and an assignment,, which decreases the variant, we define a constructive function, cg, called the co-invariant generator, which has the property thatI cg (, I) wp (, I), where wp (, I) is the weakest precondition for to establish I. Several results about the co-invariant generator are proved, important special cases are considered, and a non-trivial example of its use in deriving the body of a loop is given. We also define a function which performs a related constructive action on terms formed from binary operations. The coinvariant generator makes a useful contribution to formalising and automating a key step in program derivation.     

Conditions for the Light-Traffic Insensitivity to Loss Probability in a GI/G/m/0 Queueing System

A loss queueing system GI/G/m/0 is considered. Let a(x) be a p.d.f. of interarrival intervals. Assume that this function behaves like cx^-1 for small x. Further let B(x) be a d.f. of service time; (1/) be the mean service time. Conditions are derived for the light-traffic insensitivity of the loss probability to the form of B(x) as (/ ) 0. In particular, the condition = 1 is necessary. Estimates for the loss probability are obtained.     

Algorithms for Minimizing Response Time in Broadcast Scheduling

In this paper we study the following problem. There are n pages which clients can request at any time. The arrival times of requests for pages are known in advance. Several requests for the same page may arrive at different times. There is a server that needs to compute a good broadcast schedule. Outputting a page satisfies all outstanding requests for the page. The goal is to minimize the average waiting time of a client. This problem has recently been shown to be NP-hard. For any fixed , 0 < \le &frac;, we give a 1/-speed, polynomial time algorithm with an approximation ratio of 1/(1 – ). For example, setting = &frac; gives a 2-speed, 2-approximation algorithm. In addition, we give a 4-speed, 1-approximation algorithm improving the previous bound of 6-speed, 1-approximation algorithm.     

Single axioms for groups

We study equations of the form (=x) which are single axioms for group theory. Earlier examples of such were found by Neumann and McCune. We prove some lower bounds on the complexity of these , showing that McCune's examples are the shortest possible. We also show that no such can have only two distinct variables. We do produce an with only three distinct variables, refuting a conjecture of Neumann. Automated reasoning techniques are used both positively (searching for and verifying single axioms) and negatively (proving that various candidate (=x) hold in some nongroup and are, hence, not single axioms).     

Automatic surface reconstruction with alpha-shape method

In this paper, we present an algorithm for the reconstruction of piecewise linear surfaces from unorganized sample points with an improved -shape. Alpha-shape provides a nice mathematical definition of the shape of a set of points, but the selection of an value is tricky in surface reconstruction. F or solving this problem and the non-uniform distribution, we scale the ball according to the points density. The method discussed in this paper might be applied for surface reconstruction, and the process is fully automatic.     

Visualizing Newton's method on fractional exponents

Newton's method for finding complex solutions of the equationz –1=0 is investigated for real values of . The bifurcations that occur, as the power varies are illuminated with computer graphics. In particular, the appearance of an attracting 2-cycle just below even powers is investigated.     

Optimal α-β trees with capacity constraint

Summary We consider a specific kind of binary trees with weighted edges. Each right edge has weight while each left edge has weight . Furthermore, no path in the tree is allowed to contain L or more consecutive -edges, where L 1 is fixed. Given, , , L and the number of nodes n, an optimal tree is one which minimizes the total weighted path length. Algorithms for constructing an optimal tree as well as all optimal trees for given , , L and n are proposed and analyzed. Timing and storage requirements are also discussed.     

Fuzzy measurement in the Mishnah and the Talmud

I discuss the attitude of Jewish law sources from the 2nd–:5th centuries to the imprecision of measurement. I review a problem that the Talmud refers to, somewhat obscurely, as impossible reduction. This problem arises when a legal rule specifies an object by referring to a maximized (or minimized) measurement function, e.g., when a rule applies to the largest part of a divided whole, or to the first incidence that occurs, etc. A problem that is often mentioned is whether there might be hypothetical situations involving more than one maximal (or minimal) value of the relevant measurement and, given such situations, what is the pertinent legal rule. Presumption of simultaneous occurrences or equally measured values are also a source of embarrassment to modern legal systems, in situations exemplified in the paper, where law determines a preference based on measured values. I contend that the Talmudic sources discussing the problem of impossible reduction were guided by primitive insights compatible with fuzzy logic presentation of the inevitable uncertainty involved in measurement. I maintain that fuzzy models of data are compatible with a positivistic epistemology, which refuses to assume any precision in the extra-conscious world that may not be captured by observation and measurement. I therefore propose this view as the preferred interpretation of the Talmudic notion of impossible reduction. Attributing a fuzzy world view to the Talmudic authorities is meant not only to increase our understanding of the Talmud but, in so doing, also to demonstrate that fuzzy notions are entrenched in our practical reasoning. If Talmudic sages did indeed conceive the results of measurements in terms of fuzzy numbers, then equality between the results of measurements had to be more complicated than crisp equations. The problem of impossible reduction could lie in fuzzy sets with an empty core or whose membership functions were only partly congruent. Reduction is impossible may thus be reconstructed as there is no core to the intersection of two measures. I describe Dirichlet maps for fuzzy measurements of distance as a rough partition of the universe, where for any region A there may be a non-empty set of - _A (upper approximation minus lower approximation), where the problem of impossible reduction applies. This model may easily be combined with probabilistic extention. The possibility of adopting practical decision standards based on -cuts (and therefore applying interval analysis to fuzzy equations) is discussed in this context. I propose to characterize the uncertainty that was presumably capped by the old sages as U-uncertainty, defined, for a non-empty fuzzy set A on the set of real numbers, whose -cuts are intervals of real numbers, as U(A) = 1/h(A) ₀ ^h(A) log [1+(A)]d, where h(A) is the largest membership value obtained by any element of A and (A) is the measure of the -cut of A defined by the Lebesge integral of its characteristic function.     

Use of potential functions in 3D rendering of fractal images from complex functions

Computer graphics is important in developing fractal images visualizing the Mandelbrot and Julia sets from a complex function. Computer rendering is a central tool for obtaining nice fractal images. We render 3D objects with the height of each complex point of a fractal image considering the diverging speed of its orbit. A potential function helps approximate this speed. We propose a new method for estimating the normal vector at the surface points given by a potential function. We consider two families of functions that exhibit interesting fractal images in a bounded region: a power function,f _{, c}(z)=z +c, where is a real number, and the Newton form of an equation, where ¦¦=1 and >0.     

On the Characteristics of Growing Cell Structures (GCS) Neural Network

In this paper, a self-developing neural network model, namely the Growing Cell Structures (GCS) is characterized. In GCS each node (or cell) is associated with a local resource counter (t). We show that GCS has the conservation property by which the summation of all resource counters always equals , thereby s is the increment added to (t) of the wining node after each input presentation and (0 < < 1.0) is the forgetting (i.e., decay) factor applied to (t) of non-wining nodes. The conservation property provides an insight into how GCS can maximize information entropy. The property is also employed to unveil the chain-reaction effect and race-condition which can greatly influence the performance of GCS. We show that GCS can perform better in terms of equi-probable criterion if the resource counters are decayed by a smaller .     

Functional Estimation with Respect to a Threshold Parameter via Dynamic Split-and-Merge

We consider a class of stochastic models for which the performance measure is defined as a mathematical expectation that depends on a parameter , say (), and we are interested in constructing estimators of in functional form (i.e., entire functions of ), which can be computed from a single simulation experiment. We focus on the case where is a continuous parameter, and also consider estimation of the derivative (). One approach for doing that, when is a parameter of the probability law that governs the system, is based on the use of likelihood ratios and score functions. In this paper, we study a different approach, called split-and-merge, for the case where is a threshold parameter. This approach can be viewed as a practical way of running parallel simulations at an infinite number of values of , with common random numbers. We give several examples showing how different kinds of parameters such as the arrival rate in a queue, the probability that an arriving customer be of a given type, a scale parameter of a service time distribution, and so on, can be turned into threshold parameters. We also discuss implementation issues.     

Computing with Contexts

We investigate a representation of contexts, expressions with holes in them, that enables them to be meaningfully transformed, in particular -converted and -reduced. In particular we generalize the set of -expressions to include holes, and on these generalized entities define -reduction (i.e., substitution) and filling so that these operations preserve -congruence and commute. The theory is then applied to allow the encoding of reduction systems and operational semantics of call-by-value calculi enriched with control, imperative and concurrent features.     

Circuit-Switched Broadcasting in Multi-Port Multi-Dimensional Torus Networks

The one-to-all broadcast is the most primary collective communication pattern in a multicomputer network. This paper studies this problem in a circuit-switched torus with -port capability, where a node can simultaneously send and receive messages at one time. This is a generalization of the one-port and all-port models. We show how to efficiently perform broadcast in tori of any dimension, any size, square or nonsquare, using near optimal numbers of steps. The main techniques used are: (i) a span-by-dimension approach, which makes our solution scalable to torus dimensions, and (ii) a squeeze-then-expand approach, which makes possible solving the difficult cases where tori are non-square. Existing results, as compared to ours, can only solve very restricted sizes or dimensions of tori, or use more numbers of steps.     

A priori Probability That Two Qubits Are Unentangled

In a previous study (P. B. Slater, Eur. Phys. J. B. 17, 471 (2000)), several remarkably simple exact results were found, in certain specialized m-dimensional scenarios (m 4), for the a priori probability that a pair of qubits is unentangled/separable. The measure used was the volume element of the Bures metric (identically one-fourth the statistical distinguishability [SD] metric). Here, making use of a newly-developed (Euler angle) parameterization of the 4 × 4 density matrices of Tilma, Byrd and Sudarshan, we extend the analysis to the complete 15-dimensional convex set (C) of arbitrarily paired qubits—the total SD volume of which is known to be ⁸ / 1680 = ⁸/2⁴ 3 5 7 5.64794. Using advanced quasi-Monte Carlo procedures (scrambled Halton sequences) for numerical integration in this high-dimensional space, we approximately (5.64851) reproduce that value, while obtaining an estimate of 0.416302 for the SD volume of separable states. We conjecture that this is but an approximation to ⁶/2310 = ⁶ / (2 3 5 7 11) 0.416186. The ratio of the two volumes, 8/11²2 .0736881, would then constitute the exact Bures/SD probability of separability. The SD area of the 14-dimensional boundary of C is 142⁷/12285 = 2 71⁷/3³ 5 7 13 34.911, while we obtain a numerical estimate of 1.75414 for the SD area of the boundary of separable states. PACS: 03.67.-; 03.65.Ud; 02.60.Jh; 02.40.Ky     

Differential geometry of a parametric family of invertible linear systems—Riemannian metric,dual affine connections,and divergence

A parametric model of systems is regarded as a geometric manifold imbedded in the enveloping manifold consisting of all the linear systems. The present paper aims at establishing a new geometrical method and framework for analyzing properties of manifolds of systems. A Riemannian metric and a pair of dual affine connections are introduced to a system manifold. They are called-connections. The duality of connections is a new concept in differential geometry. The manifold of all the linear systems is-flat so that it admits natural and invariant-divergence measures. Such geometric structures are useful for treating the problems of approximation, identification, and stochastic realization of systems. By using the-divergences, we solve the problem of approximating a given system by one included in a model. For a sequence of-flat nesting models such as AR models and MA models, it is shown that the approximation errors are decomposed additively corresponding to each dimension of the model.     

Maintaining the visibility map of spheres while moving the viewpoint on a circle at infinity

We investigate three-dimensional visibility problems for scenes that consist ofn non-intersecting spheres. The viewing point moves on a flightpath that is part of a circle at infinity given by a planeP and a range of angles {(t)¦t[01]} [02]. At timet, the lines of sight are parallel to the ray inP, which starts in the origin ofP and represents the angle(t) (orthographic views of the scene). We give an algorithm that computes the visibility graph at the start of the flight, all time parameters at which the topology of the scene changes, and the corresponding topology changes. The algorithm has running time0(n + k + p) logn), wheren is the number of spheres in the scene;p is the number of transparent topology changes (the number of different scene topologies visible along the flight path, assuming that all spheres are transparent); andk denotes the number of vertices (conflicts) which are in the (transparent) visibility graph at the start and do not disappear during the flight.The second author was supported by the ESPRIT II Basic Research Actions Program, under Contract No. 3075 (project ALCOM).     

Mathematical Analysis and Optimization of Infiltration Processes

A variety of infiltration techniques can be used to fabricate solid materials, particularly composites. In general these processes can be described with at least one time dependent partial differential equation describing the evolution of the solid phase, coupled to one or more partial differential equations describing mass transport through a porous structure. This paper presents a detailed mathematical analysis of a relatively simple set of equations which is used to describe chemical vapor infiltration. The results demonstrate that the process is controlled by only two parameters, and . The optimization problem associated with minimizing the infiltration time is also considered. Allowing and to vary with time leads to significant reductions in the infiltration time, compared with the conventional case where and are treated as constants.