A STUDY OF PENALTY FORMULATIONS USED IN THE NUMERICAL APPROXIMATION OF A RADIALLY SYMMETRIC ELASTICITY PROBLEM

We consider a class of two-dimensional problems in classical linear elasticity for which material overlapping occurs in the absence of singularities. Of course, material overlapping is not physically realistic, and one possible way to prevent it uses a constrained minimization theory. In this theory, a minimization problem consists of minimizing the total potential energy of a linear elastic body subject to the constraint that the deformation field must be locally invertible. Here, we use an interior and an exterior penalty formulation of the minimization problem together with both a standard finite element method and classical nonlinear programming techniques to compute the minimizers. We compare both formulations by solving a plane problem numerically in the context of the constrained minimization theory. The problem has a closed-form solution, which is used to validate the numerical results. This solution is regular everywhere, including the boundary. In particular, we show numerical results which indicate that, for a fixed finite element mesh, the sequences of numerical solutions obtained with both the interior and the exterior penalty formulations converge to the same limit function as the penalization is enforced. This limit function yields an approximate deformation field to the plane problem that is locally invertible at all points in the domain. As the mesh is refined, this field converges to the exact solution of the plane problem.

A fast two-grid and finite section method for a class of integral equations on the real line with application to an acoustic scattering problem in the half-plane

We consider the numerical treatment of second kind integral equations on the real line of the form ∅(s) = ∫_(-∞)^(+∞)▒〖κ(s-t)z(t)ϕ(t)dt,s=R〗 (abbreviated ϕ= ψ+K_z ϕ) in which K ϵ L_1 (R), z ϵ L_∞ (R) and ψ ϵ BC(R), the space of bounded continuous functions on R, are assumed known and ϕ ϵ BC(R) is to be determined. We first derive sharp error estimates for the finite section approximation (reducing the range of integration to [-A, A]) via bounds on (1-K_z )^(-1)as an operator on spaces of weighted continuous functions. Numerical solution by a simple discrete collocation method on a uniform grid on R is then analysed: in the case when z is compactly supported this leads to a coefficient matrix which allows a rapid matrix-vector multiply via the FFT. To utilise this possibility we propose a modified two-grid iteration, a feature of which is that the coarse grid matrix is approximated by a banded matrix, and analyse convergence and computational cost. In cases where z is not compactly supported a combined finite section and two-grid algorithm can be applied and we extend the analysis to this case. As an application we consider acoustic scattering in the half-plane with a Robin or impedance boundary condition which we formulate as a boundary integral equation of the class studied. Our final result is that if z (related to the boundary impedance in the application) takes values in an appropriate compact subset Q of the complex plane, then the difference between ϕ(s)and its finite section approximation computed numerically using the iterative scheme proposed is ≤C_1 [kh log⁡〖(1⁄kh)+(1-Θ)^((-1)⁄2) (kA)^((-1)⁄2) 〗 ] in the interval [-ΘA,ΘA](Θ<1) for kh sufficiently small, where k is the wavenumber and h the grid spacing. Moreover this numerical approximation can be computed in ≤C_2 N log⁡N operations, where N = 2A/h is the number of degrees of freedom. The values of the constants C1 and C2 depend only on the set Q and not on the wavenumber k or the support of z.

The impedance boundary value problem for the Helmholtz equation in a half-plane

We prove unique existence of solution for the impedance (or third) boundary value problem for the Helmholtz equation in a half-plane with arbitrary L∞ boundary data. This problem is of interest as a model of outdoor sound propagation over inhomogeneous flat terrain and as a model of rough surface scattering. To formulate the problem and prove uniqueness of solution we introduce a novel radiation condition, a generalization of that used in plane wave scattering by one-dimensional diffraction gratings. To prove existence of solution and a limiting absorption principle we first reformulate the problem as an equivalent second kind boundary integral equation to which we apply a form of Fredholm alternative, utilizing recent results on the solvability of integral equations on the real line in [5].

Scattering by rough surfaces: the Dirichlet problem for the Helmholtz equation in a non-locally perturbed half-plane

We consider the Dirichlet boundary value problem for the Helmholtz equation in a non-locally perturbed half-plane, this problem arising in electromagnetic scattering by one-dimensional rough, perfectly conducting surfaces. We propose a new boundary integral equation formulation for this problem, utilizing the Green's function for an impedance half-plane in place of the standard fundamental solution. We show, at least for surfaces not differing too much from the flat boundary, that the integral equation is uniquely solvable in the space of bounded and continuous functions, and hence that, for a variety of incident fields including an incident plane wave, the boundary value problem for the scattered field has a unique solution satisfying the limiting absorption principle. Finally, a result of continuous dependence of the solution on the boundary shape is obtained.

The rolling ball problem on the plane revisited

By a sequence of rollings without slipping or twisting along segments of a straight line of the plane, a spherical ball of unit radius has to be transferred from an initial state to an arbitrary final state taking into account the orientation of the ball. We provide a new proof that with at most 3 moves, we can go from a given initial state to an arbitrary final state. The first proof of this result is due to Hammersley ( 1983). His proof is more algebraic than ours which is more geometric. We also showed that generically no one of the three moves, in any elimination of the spin discrepancy, may have length equal to an integral multiple of 2 pi.

A smooth evolutionary structural optimization procedure applied to plane stress problem

Topological optimization problems based on stress criteria are solved using two techniques in this paper. The first technique is the conventional Evolutionary Structural Optimization (ESO), which is known as hard kill, because the material is discretely removed; that is, the elements under low stress that are being inefficiently utilized have their constitutive matrix has suddenly reduced. The second technique, proposed in a previous paper, is a variant of the ESO procedure and is called Smooth ESO (SESO), which is based on the philosophy that if an element is not really necessary for the structure, its contribution to the structural stiffness will gradually diminish until it no longer influences the structure; its removal is thus performed smoothly. This procedure is known as "soft-kill"; that is, not all of the elements removed from the structure using the ESO criterion are discarded. Thus, the elements returned to the structure must provide a good conditioning system that will be resolved in the next iteration, and they are considered important to the optimization process. To evaluate elasticity problems numerically, finite element analysis is applied, but instead of using conventional quadrilateral finite elements, a plane-stress triangular finite element was implemented with high-order modes for solving complex geometric problems. A number of typical examples demonstrate that the proposed approach is effective for solving problems of bi-dimensional elasticity. (C) 2014 Elsevier Ltd. All rights reserved.

Tight frames of k-plane ridgelets and the problem of representing objects that are smooth away from d-dimensional singularities in Rn

For each pair (n, k) with 1 ≤ k < n, we construct a tight frame (ρλ : λ ∈ Λ) for L2 (Rn), which we call a frame of k-plane ridgelets. The intent is to efficiently represent functions that are smooth away from singularities along k-planes in Rn. We also develop tools to help decide whether k-plane ridgelets provide the desired efficient representation. We first construct a wavelet-like tight frame on the X-ray bundle χn,k—the fiber bundle having the Grassman manifold Gn,k of k-planes in Rn for base space, and for fibers the orthocomplements of those planes. This wavelet-like tight frame is the pushout to χn,k, via the smooth local coordinates of Gn,k, of an orthonormal basis of tensor Meyer wavelets on Euclidean space Rk(n−k) × Rn−k. We then use the X-ray isometry [Solmon, D. C. (1976) J. Math. Anal. Appl. 56, 61–83] to map this tight frame isometrically to a tight frame for L2(Rn)—the k-plane ridgelets. This construction makes analysis of a function f ∈ L2(Rn) by k-plane ridgelets identical to the analysis of the k-plane X-ray transform of f by an appropriate wavelet-like system for χn,k. As wavelets are typically effective at representing point singularities, it may be expected that these new systems will be effective at representing objects whose k-plane X-ray transform has a point singularity. Objects with discontinuities across hyperplanes are of this form, for k = n − 1.

On the numerical solution of a Cauchy problem in an elastostatic half-plane with a bounded inclusion

We propose an iterative procedure for the inverse problem of determining the displacement vector on the boundary of a bounded planar inclusion given the displacement and stress fields on an infinite (planar) line-segment. At each iteration step mixed boundary value problems in an elastostatic half-plane containing the bounded inclusion are solved. For efficient numerical implementation of the procedure these mixed problems are reduced to integral equations over the bounded inclusion. Well-posedness and numerical solution of these boundary integral equations are presented, and a proof of convergence of the procedure for the inverse problem to the original solution is given. Numerical investigations are presented both for the direct and inverse problems, and these results show in particular that the displacement vector on the boundary of the inclusion can be found in an accurate and stable way with small computational cost.

Magnetoconfined levels in a parabolic quantum dot: An analytical solution of a three-dimensional Fock-Darwin problem in a tilted magnetic field

The energy spectrum of an electron confined in a quantum dot (QD) with a three-dimensional anisotropic parabolic potential in a tilted magnetic field was found analytically. The theory describes exactly the mixing of in-plane and out-of-plane motions of an electron caused by a tilted magnetic field, which could be seen, for example, in the level anticrossing. For charged QDs in a tilted magnetic field we predict three strong resonant lines in the far-infrared-absorption spectra.

A contribution to the exact modal solution of in-plane beam structures

The exact vibration modes and natural frequencies of planar structures and mechanisms, comprised Euler-Bernoulli beams, are obtained by solving a transcendental. nonlinear, eigenvalue problem stated by the dynamic stiffness matrix (DSM). To solve this kind of problem, the most employed technique is the Wittrick-Williams algorithm, developed in the early seventies. By formulating a new type of eigenvalue problem, which preserves the internal degrees-of-freedom for all members in the model, the present study offers an alternative to the use of this algorithm. The new proposed eigenvalue problem presents no poles, so the roots of the problem can be found by any suitable iterative numerical method. By avoiding a standard formulation for the DSM, the local mode shapes are directly calculated and any extension to the beam theory can be easily incorporated. It is shown that the method here adopted leads to exact solutions, as confirmed by various examples. Extensions of the formulation are also given, where rotary inertia, end release, skewed edges and rigid offsets are all included. (C) 2008 Elsevier Ltd. All rights reserved.

Computation of bifurcation boundaries for power systems: A new Delta-plane method

This paper is devoted to the problems of finding the load flow feasibility, saddle node, and Hopf bifurcation boundaries in the space of power system parameters. The first part contains a review of the existing relevant approaches including not-so-well-known contributions from Russia. The second part presents a new robust method for finding the power system load flow feasibility boundary on the plane defined by any three vectors of dependent variables (nodal voltages), called the Delta plane. The method exploits some quadratic and linear properties of the load now equations and state matrices written in rectangular coordinates. An advantage of the method is that it does not require an iterative solution of nonlinear equations (except the eigenvalue problem). In addition to benefits for visualization, the method is a useful tool for topological studies of power system multiple solution structures and stability domains. Although the power system application is developed, the method can be equally efficient for any quadratic algebraic problem.

Factors associated with pelvic asymmetry in transverse plane during gait in patients with cerebral palsy

The purpose of this study was to describe the patterns of pelvic rotational asymmetry in the transverse plane and identify the possible factors related to this problem. One thousand and forty-five patients with cerebral palsy (CP) and complete documentation in the gait laboratory were reviewed in a retrospective study. Pelvic asymmetry in the transverse plane was observed in 52.7% of the patients; and to identify the possible causes of pelvic retraction, clinical (Thomas test, popliteal angle, and gastrocnemius tightness) and dynamic parameters (mean rotation of the hip in stance, minimum hip flexion, minimum knee flexion, and peak ankle dorsiflexion) were evaluated. The association between these parameters and pelvic retraction was assessed statistically. The results showed that 75.7% of patients with asymmetric pattern of the pelvis had clinical diagnosis of diplegic spastic CP. Among the patients with asymmetrical CP, the most common pattern was pelvic retraction on the affected side. The relationship between pelvic retraction and internal hip rotation was stronger in patients with asymmetrical diplegic CP than in those with hemiplegic (P<0.001) or symmetrical diplegic CP (P=0.014). All of the patients exhibited a significant association among clinical parameters (Thomas test, popliteal angle, and gastrocnemius tightness) and pelvic retraction. In conclusion, pelvic retraction seems to be a multifactorial problem, and the etiology can change according to topographic classification, which must be taken into account during the decision-making process in patients with CP. J Pediatr Orthop B 18:320-324 (C) 2009 Wolters Kluwer Health vertical bar Lippincott Williams & Wilkins.

Plane-to-plane positioning from image-based visual servoing and structured light

In this paper we face the problem of positioning a camera attached to the end-effector of a robotic manipulator so that it gets parallel to a planar object. Such problem has been treated for a long time in visual servoing. Our approach is based on linking to the camera several laser pointers so that its configuration is aimed to produce a suitable set of visual features. The aim of using structured light is not only for easing the image processing and to allow low-textured objects to be treated, but also for producing a control scheme with nice properties like decoupling, stability, well conditioning and good camera trajectory

Driver scheduling problem modelling

The Drivers Scheduling Problem (DSP) consists of selecting a set of duties for vehicle drivers, for example buses, trains, plane or boat drivers or pilots, for the transportation of passengers or goods. This is a complex problem because it involves several constraints related to labour and company rules and can also present different evaluation criteria and objectives. Being able to develop an adequate model for this problem that can represent the real problem as close as possible is an important research area.The main objective of this research work is to present new mathematical models to the DSP problem that represent all the complexity of the drivers scheduling problem, and also demonstrate that the solutions of these models can be easily implemented in real situations. This issue has been recognized by several authors and as important problem in Public Transportation. The most well-known and general formulation for the DSP is a Set Partition/Set Covering Model (SPP/SCP). However, to a large extend these models simplify some of the specific business aspects and issues of real problems. This makes it difficult to use these models as automatic planning systems because the schedules obtained must be modified manually to be implemented in real situations. Based on extensive passenger transportation experience in bus companies in Portugal, we propose new alternative models to formulate the DSP problem. These models are also based on Set Partitioning/Covering Models; however, they take into account the bus operator issues and the perspective opinions and environment of the user.We follow the steps of the Operations Research Methodology which consist of: Identify the Problem; Understand the System; Formulate a Mathematical Model; Verify the Model; Select the Best Alternative; Present the Results of theAnalysis and Implement and Evaluate. All the processes are done with close participation and involvement of the final users from different transportation companies. The planner s opinion and main criticisms are used to improve the proposed model in a continuous enrichment process. The final objective is to have a model that can be incorporated into an information system to be used as an automatic tool to produce driver schedules. Therefore, the criteria for evaluating the models is the capacity to generate real and useful schedules that can be implemented without many manual adjustments or modifications. We have considered the following as measures of the quality of the model: simplicity, solution quality and applicability. We tested the alternative models with a set of real data obtained from several different transportation companies and analyzed the optimal schedules obtained with respect to the applicability of the solution to the real situation. To do this, the schedules were analyzed by the planners to determine their quality and applicability. The main result of this work is the proposition of new mathematical models for the DSP that better represent the realities of the passenger transportation operators and lead to better schedules that can be implemented directly in real situations.

Particle creation in a colliding plane wave spacetime: Wave packet quantization

We use wave packet mode quantization to compute the creation of massless scalar quantum particles in a colliding plane wave spacetime. The background spacetime represents the collision of two gravitational shock waves followed by trailing gravitational radiation which focus into a Killing-Cauchy horizon. The use of wave packet modes simplifies the problem of mode propagation through the different spacetime regions which was previously studied with the use of monochromatic modes. It is found that the number of particles created in a given wave packet mode has a thermal spectrum with a temperature which is inversely proportional to the focusing time of the plane waves and which depends on the mode trajectory.