Geometrically exact solution of a buckling column with asymmetric boundary conditions

For the symmetrically supported Euler buckling column with both ends hinged the classical stability theory yields simple trigonometric functions as buckling modes, i.e. w(x) = A sin αx. The eigenvalues α are just multiples of π. In comparison, the analysis of the asymmetrically supported Euler buckling column with one end clamped and the other end hinged is more complicated: The buckling modes are a combination of trigonometric functions in form of w(x) = A (sin αx − αx cos (αL)). The eigenvalues follow from a transcendental equation. Applying a geometrically exact theory to the aforementioned Euler buckling problems, a similar relation in the complexity of the analyses will naturally arise. Using, e.g., the elastica model the buckling behavior of the symmetrically supported column is represented by elliptic integrals. However, the determination of the buckling behavior of the asymmetrically supported column turns out to be much more complex and elaborate. This article presents a direct comparison of the symmetrically and asymmetrically supported buckling columns regarding their analyses by means of classical stability theory and by the geometrically exact theory of the elastica. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Application of Rothe’s Method to a Parabolic Inverse Problem with Nonlocal Boundary Condition

We consider an inverse problem for the determination of a purely time-dependent source in a semilinear parabolic equation with a nonlocal boundary condition. An approximation scheme for the solution together with the well-posedness of the problem with the initial value u₀ ∈ H¹(Ω) is presented by means of the Rothe time-discretization method. Further approximation scheme via Rothe’s method is constructed for the problem when u₀ ∈ L²(Ω) and the integral kernel in the nonlocal boundary condition is symmetric.

     

The Inverse Problem of Simultaneous Determination of the Two Time-Dependent Lower Coefficients in a Nondivergent Parabolic Equation in the Plane

We prove existence and uniqueness theorems for the solution of the inverse problem of simultaneous determination of the t-dependent coefficients of u and u_x in a nondivergent parabolic equation with two independent variables from integral observation of x. Estimates of the maxima of the moduli of these coefficients with constants explicitly expressed in terms of the input data of the problem are given. An example of an inverse problem to which the proved theorems apply is presented.

     

Solving the Inverse Generalized Problem of Vertical Seismic Profiling

The dynamic inverse seismics problem is considered in a generalized setting. We investigate whether the wave propagation problem in a vertically nonhomogeneous medium is well-posed. We show that the regular part of the solution is an L ₂ function and the inverse problem, i.e., the determination of the reflection coefficient, is thus reducible to minimizing the error functional. The gradient of the functional is obtained in explicit form from the conjugate problem, and approximate formulas for its evaluation are derived. A regularization algorithm for the solution of the inverse problem is considered; simulation results using various excitation sources are reported.     

Buckling for an elastoplastic plate with an incremental constitutive relation

We study the buckling of an elastoplastic plate loaded on its edges by a planar thrust(t)F(x ₁ ,x ₂ ) depending on time. We consider an incremental constitutive relation which permits the distinction between loading and unloading. This could be obtained by taking a Prandtl-Reuss relation, but, for technical reasons, we restrict ourselves to an approximation of this relation. We give a necessary condition for the existence of a nontrivial solution (together with the zero one) after a timet ₀ ; an important feature is that this condition takes into account the whole story of the loading and cannot be expressed in terms of its value at a single time. We prove the existence of bifurcated branches, for some sufficient condition involving the slope of at timet ₀ , and describe their local behavior; we point out the fact that buckling can actually occur when the loading is decreasing.Part of this work was done while the authors were visiting the I.M.A. in Minneapolis, Minnesota, USA.     

Unique solvability of the inverse problem of determination of the leading coefficient in a parabolic equation

We study existence and uniqueness of the solution for the inverse problem of determination of the unknown coefficient ϱ(t) multiplying u _t in a nondivergence parabolic equation. As additional information, the integral of the solution over the domain of space variables with some given weight function is specified. The coefficients of the equation depend both on time and on the space variables.     

Inverse problems in underwater acoustics and approximation of the velocity field for signal propagation

A new approach is suggested for solving the inverse problem in underwater acoustics — the determination of the velocity of signal propagation in an inhomogeneous medium. The unknown velocity u(x(t), t) is assumed to be independent of the time t and is sought in the form of a polynomial with respect to the degree of the space coordinate x ₁, x₂, x₃.Translated fromVychislitel'naya i Prikladnaya Matematika, No. 69, pp. 102–105, 1989.     

Ein gemischtes randwertproblem der kreiszylinderschale mit beliebigen ausschnitten

The problem of stress determination in the area of cut-outs in circular cylindrical shells at given loads is of great interest in industrial practice. This work deals with a mixed boundary value problem of a differential equation derived according to the theory of shallow shells. On part C_t1 of the boundary, the displacements are given, whereas the stresses are specified on the remaining part C_t2. Starting from the Betti-Maxwell principle and with the aid of the fundamental solutions for unit loads and unit displacements, integral representations can be derived for the displacement functions as well as the stress functions. The problem is then transformed into an equivalent system of Fredholm integral equations of the first kind with logarithmic kernels as the main part. As the integral equations together with the auxiliary conditions form a strongly elliptical system of pseudo-differential operators, the Galerkin method converges. Assuming that curves C_t1 and C_t1 do not have points of intersection and that the data are sufficiently regular, the required functions are approximated by cubic splines and, for simplicity's sake, the integral equation system is solved by approximation with a collocation method. In view of the complicated terms of the kernel functions, the kernels are split into a regular and a singular part, the regular part being in turn replaced by cubic splines. The remaining integrations are done numerically by means of Gaussian quadrature formulae. The applicability of the method is demonstrated with the example of a cylinder under internal pressure.     

On surface area measures of convex bodies

The set L _j of jth-order surface area measures of convex bodies in d-space is well known for j=d–1. A characterization of L _j was obtained by Firey and Berg. The determination of L _j, for j{2, ..., d–2}, is an open problem. Here we show some properties of L _j concerning convexity, closeness, and size. Especially we prove that the difference set L _j–L _j is dense (in the weak topology) in the set of signed Borel measures on the unit sphere which have barycentre 0.     

Approximation of the conductivity coefficient in the heat equation

In this paper, we study the conductivity coefficient determination in the heat equation from observation of the lateral Dirichlet-to-Neumann map. We define a bilinear form function Q_γ associated to the boundary condition and the Dirichlet-to-Neumann map, and prove that the linearized problem d?Q_γ is injective. Based on the idea of complex geometrical optics solutions, we give two approximations to the conductivity coefficient by using the Fourier truncation method and the mollification method. Under the a priori assumption of the conductivity, we estimate the errors between the conductivity coefficient and its approximations by setting a suitable bound of the frequency.     

Non-commutative spaces with transitive translation groups. II

LetG be a group. The problem of finding all skewaffine spaces with a transitive translation group isomorphic toG will be reduced to the determination of all Schur-decompositions ofG, i.e. of all decompositions ofG\{1} into pairwise disjoint subsetsA _r such that anyA _r ^–1 is anA _s and any productA _r A _s is a union of someA _t and possibly {1}.Dedicated to Helmut Salzmann on his 65th birthday     

General stability of composite panels reinforced with flexible rods taking account of the side boundary conditions

Reinforced panels are the basic load-bearing elements of various structures. Optimization of massive structures requires consideration of deformation of the panel cross-sections. This is particularly important in determining the bearing strength at buckling. The load scheme, conditions for fixation of the panel cross-section, and bend-torsional stiffness taking account of the deformation of the rod cross-section affect the buckling load in real structures. The stress distribution prior to buckling must be known to solve the buckling problem properly. The stress in the panel is proportional to the active load. The stress distribution is assumed to be known according to our previous method [1]. The load scheme and panel dimensions are shown in Fig. 1. The stress distribution in the panel prior to buckling can be found using Eqs. (1)-(3). A view of the cross-section is given in Fig. 1. The displacements in the panel at buckling for the boundary area are found using Eqs. (4)-(6), while the stresses in the skin and stiffness are found using Eq. (7). Roots k₁ and k₂ are those of the characteristic equation and is a dimensionless coordinate. The problem was solved using variational theory. The potential energy is given by Eqs. (8) and (9) by orihogonalization of Eqs. (5). The basic equations are converted to Eqs. (10) by evaluation of the components in Eqs. (8) and (9). Its calculation (11) gives the compression load. Optimization of parameter gives the critical strength P₁ = 6.93 kN (without taking account of the boundary area) and P₂ = 5.31 kN (taking account of the boundary area).Translated from Mekhanika Kompozitnikh Materialov, Vol. 30, No. 4, pp. 540–546, July–August, 1994.     

Finding cheapest cycles in vertex-weighted quasi-transitive and extended semicomplete digraphs

We consider the problem of finding a minimum cost cycle in a digraph with real-valued costs on the vertices. This problem generalizes the problem of finding a longest cycle and hence is NP-hard for general digraphs. We prove that the problem is solvable in polynomial time for extended semicomplete digraphs and for quasi-transitive digraphs, thereby generalizing a number of previous results on these classes. As a byproduct of our method we develop polynomial algorithms for the following problem: Given a quasi-transitive digraph D with real-valued vertex costs, find, for each j=1,2,…,|V(D)|, j disjoint paths P₁,P₂,…,P_j such that the total cost of these paths is minimum among all collections of j disjoint paths in D.     

Relación entre el parámetro flecha-luz y la carga crítica de pandeo lateral para arcos parabólicos biempotrados bajo diferentes hipótesis de cálculo

Curved graphs that relate the f/L ratio and the λ = qL³/EI_y parameter for clamped arches with the different calculation hypothesis are going to be proposed in this article.It has been necessary to determine the real critical lateral strength of a series of clamped arches considering different calculation hypothesis respect to geometrical linearity and a steel material characteristic.The elastic-plastic material hypothesis with non geometrical linearity considers the existence of residual stresses and initial lateral deflection.     

Buckling and lateral buckling interaction in thin-walled beam-column elements with mono-symmetric cross sections

Effects of axial forces on beam lateral buckling strength are investigated here in the case of elements with mono-symmetric cross sections. A unique compact closed-form is established for the interaction of lateral buckling moment with axial forces. This new equation is derived from a non-linear stability model. It includes first order bending distribution, load height level and effect of mono-symmetry terms (Wagner’s coefficient and shear point position). Compared to the so-called three-factors (C₁–C₃) formula commonly employed in beam lateral buckling stability, another factor C₄ is added in presence of axial loads. Pre-buckling deflection effects are considered in the study and the case of doubly-symmetric cross sections is easily recovered. The proposed solutions are validated and compared to finite element simulations where 3D beam elements including warping are used. The agreement of the proposed solutions with bifurcations observed on the non-linear equilibrium paths is good. Dimensionless interaction curves are dressed for the beam lateral buckling strength and the applied axial load, where the flexural-torsional buckling axial force is a taken as reference.     

Computing the bump number with techniques from two-processor scheduling

Let (X, <) be a partially ordered set. A linear extension x ₁, x ₂, ... has a bump whenever x _i<x _i+1, and it has a jump whenever x _iand x _i+1are incomparable. The problem of finding a linear erxtension that minimizes the number of jumps has been studied extensively; Pulleyblank shows that it is NP-complete in the general case. Fishburn and Gehrlein raise the question of finding a linear extension that minimizes the number of bumps. We show that the bump number problem is closely related to the well-studied problem of scheduling unit-time tasks with a precedence partial order on two identical processors. We point out that a variant of Gabow's linear-time algorithm for the two-processor scheduling problem solves the bump number problem. Habib, Möhring, and Steiner have independently discovered a different polynomial-time algorithm to solve the bump number problem.Part of this work was done while the first author was a Research Student Associate at IBM Almaden Research Center. During the academic year his work is primarily supported by a Fannie and John Hertz Foundation Fellowship and is supported in part by ONR contract N00014-85-C-0731.     

Energetic reasoning and bin-packing problem, for bounding a parallel machine scheduling problem

This paper deals with an extension of energetic reasoning, using some efficient lower bounds of the bin-packing problem, to get tight lower bounds for the P|r _i , q _i |C _max. The link between P||C _max and bin-packing problem is well-known. Our purpose is to extend the use of efficient lower bounds of the bin-packing problem to P|r _i , q _i |C _max. We focus on some time-intervals, to compute the mandatory parts of activities within this time-interval and then to deduce an associated bin-packing instance. Thus, lower bounds of the bin-packing problem are used to get new satisfiability tests for the parallel machine problem. We also propose to extend the classical time-bound adjustments of release dates and deadlines to efficiently use bin-packing lower bounds. Experimental results that prove the efficiency of our approach on several kind of instances are reported.     

Induced disjoint paths problem in a planar digraph

As an extension of the disjoint paths problem, we introduce a new problem which we call the induced disjoint paths problem. In this problem we are given a graph G and a collection of vertex pairs {(s₁,t₁),…,(s_k,t_k)}. The objective is to find k paths P₁,…,P_k such that P_i is a path from s_i to t_i and P_i and P_j have neither common vertices nor adjacent vertices for any distinct i,j.The induced disjoint paths problem has several variants depending on whether k is a fixed constant or a part of the input, whether the graph is directed or undirected, and whether the graph is planar or not. We investigate the computational complexity of several variants of the induced disjoint paths problem. We show that the induced disjoint paths problem is (i) solvable in polynomial time when k is fixed and G is a directed (or undirected) planar graph, (ii) NP-hard when k=2 and G is an acyclic directed graph, (iii) NP-hard when k=2 and G is an undirected general graph.As an application of our first result, we show that we can find in polynomial time certain structures called a “hole” and a “theta” in a planar graph.     

A Globally Convergent Algorithm for a PDE-Constrained Optimization Problem Arising in Electrical Impedance Tomography

We study the convergence properties of an algorithm for the inverse problem of electrical impedance tomography, which can be reduced to a partial differential equation (PDE) constrained optimization problem. The direct problem consists of the potential equation div(??u) = 0 in a circle, with Neumann condition describing the behavior of the electrostatic potential in a medium with conductivity given by the function ?(x, y). We suppose that at each time a current ψ _i is applied to the boundary of the circle (Neumann's data), and that it is possible to measure the corresponding potential ? _i (Dirichlet data). The inverse problem is to find ?(x, y) given a finite number of Cauchy pairs measurements (? _i , ψ _i ), i = 1,…, N. The problem is formulated as a least squares problem, and the developed algorithm solves the continuous problem using descent iterations in its corresponding finite element approximations. Wolfe's conditions are used to ensure the global convergence of the optimization algorithm for the continuous problem. Although exact data are assumed, measurement errors in data and regularization methods shall be considered in a future work.