On Accuracy Problems for Semi-Analytical Sensitivity Analyses

Abstract

The semi-analytical method of sensitivity analysis combines ease of implementation with computational efficiency. A major drawback to this method, however, is that severe accuracy problems have recently been reported. A complete error analysis for a beam problem with changing length is carried out in this paper. It is shown that the sensitivity error is proportional to the relative length difference, but in agreement with Eq. 3.8. The approximate pseudo loads thus violate moment equilibrium for rigid body motion while the correct pseudo loads do not.

It might be a good idea to modify the approximate pseudo loads in order to obtain general load equilibrium with rigid body motions. Such a method would be readily applicable for any element type, whether analytical expressions for the element stiffnesses are available or not. This topic is postponed for a future study.     

A Global,Direct Algorithm for Path-Following and Active Static Control of Elastic Bar Structures*

ABSTRACT

A method is presented by means of which the equilibrium path of any elastic bar structure may be traced globally, without applying iteration techniques. The basic idea is that the bar structure is reduced to a set of Initial Value Problems (IVPs) with parameters, and the equilibrium path is piecewise linearly interpolated in the parameter space. The way in which this method is capable of handling problems of active static control is demonstrated. The mathematical basis of this type of method is described by Allgower and Georg [1] as the Piecewise Linear (PL) algorithm. Here it is shown how this algorithm can be applied to problems in structural mechanics.     

On Large-Time Behavior of Solutions to the Compressible Navier-Stokes Equations in the Half Space in R 3

 The Navier-Stokes equation for compressible viscous fluid is considered on the half space in R ³ under the zero-Dirichlet boundary condition for the momentum with initial data near an arbitrarily given equilibrium of positive constant density and zero momentum. Time decay properties in L ² norms for solutions of the linearized problem are investigated to obtain the rate of convergence in L ² norms of solutions to the equilibrium when initial data are sufficiently close to the equilibrium in . Some lower bounds are derived for solutions to the linearized problem, one of which indicates a nonlinear phenomenon not appearing in the case of the Cauchy problem on the whole space. (Accepted May 8, 2002) Published online October 18, 2002 Communicated by T.-P. LIU     

Postbuckling Behavior of Thin-Walled Open Cross-Section Compression Members

ABSTRACT

The postbuckling behavior of simply supported columns with thin-walled open cross section is investigated by means of the general nonlinear theory of elastic stability. Fourth-order terms in the series expansion of the total potential energy are disregarded.

It is shown that interaction between linearly independent simultaneous buckling modes is responsible for neutral equilibrium at bifurcation if the column cross section has two axes of symmetry, and unstable if it has only one. If the buckling modes are not simultaneous, the equilibrium is neutral in both cases. Finally, the equilibrium at bifurcation is usually unstable if the cross section has no axis of symmetry.     

Shakedown Analysis of Elastoplastic Arches

Abstract

The use of mathematical programming methods proves to be of interest in the formulation and the solution of some rather tedious problems of structural plasticity. This paper extends the application of linear programming techniques to elastoplastic arches under variable repeated loadings by adopting linearized models for the arch shape and the yield criteria of its sections. The concept of M and M + N mechanisms proves convenient in formulating the equilibrium equations by the static approach. With nonlinear yield conditions the formulations presented will still hold, but the solution of corresponding nonlinear programming problems will likely become more complex. In addition to their practical applications, the linear programming approaches suggested and illustrated have the advantage of allowing the incremental collapse load to be derived by a direct, algorithmic procedure, rather than by the trial and error procedures suggested in earlier investigations.     

Unsteady Flux-Vector-Based Green Element Method

This article extends the mathematical formulation and solution procedure of the modified ‘q-based’ GEM to unsteady situations, namely to the modified unsteady ‘q-based’ GEM. Solutions that provide information on the evolution of the pressure and the flux over long time intervals are available by incorporating the additional dimension of time into steady problems. This approach is first tested by solving an example for which an analytical solution is available. The numerical results for this example is found to be in excellent agreement with the analytical solution. Several problems involving geological features, such as wells and faults, are then investigated, with different properties applying to the faults. A strong influence of the low permeability faults is in evidence in these problems.     

Optimal Elastic Design for Prescribed Maximum Deflection

ABSTRACT

For the problems of the optimal elastic design with prescribed maximum deflection, a variational formulation is proposed, with reference to the one-or two-dimensional bending structures.

Necessary optimum conditions are found, and the physical features of the optimal solutions are discussed for the “absolute” minimum cost problems, and, when dealing with beams, for the solutions with piece-wise constant design function.

Some examples are solved by using numerical methods that are directly derived from the variational formulation.     

On the adjacent equilibrium method in the stability theory of conservative elastic continua

The relationship of the adjacent equilibrium method, the regular perturbation method and the energy method for neutral equilibrium is studied. It is shown that unlike the adjacent equilibrium method, the regular perturbation method yields, for the problems under consideration, non-homogeneous perturbation equations and that adjacent states of equilibrium do not exist at the bifurcation point. These results are then compared with the result of the energy criterion for neutral equilibrium V₂[u] = 0. It is found that although the physical arguments are different in the three methods, the resulting stability equations are identical; thus showing why the adjacent equilibrium argument, even for cases when it is incorrect, yields correct critical loads. This is followed by a discussion of an incorrect derivation of a stability condition and a notion about a load type introduced in the stability literature, which are consequences of the assumption of the general existence of adjacent equilibrium states at bifurcation points.     

Integral formulation and self-consistent modelling of elastoviscoplastic behavior of heterogeneous materials

Summary Based on projection operators, an integral formulation is proposed for elastoviscoplastic heterogeneous materials. The problem requires a complete mechanical formulation, including the static equilibrium property concerning the known field σ, in addition to the classical field equations concerning the unknown fields ɛ˙ and σ˙. The formulation leads to an integral equation, in which elasticity and viscoplasticity effects interact through an homogeneous elastoviscoplastic medium with elastic moduli C and viscoplastic moduli B. To approximate the integral equation, the self-consistent scheme is followed. In order to obtain consistent approximation conditions, we introduce fluctuations of elastic and viscoplastic strain rate fields with respect to known kinematically compatible fields. It results in a strain rate concentration relation connecting the strain rate at each point to the macroscopic loading conditions and the local stress field. The results are presented and compared with other models and with experimental data in the case of a two-phase material. Received 26 August 1997; accepted for publication 2 July 1998     

Instability of equilibrium of nonholonomic systems with dissipation and circulatory forces

The paper discusses the equilibrium instability problem of the scleronomic nonholonomic systems acted upon by dissipative, conservative, and circulatory forces. The method is based on the existence of solutions to the differential equations of the motion which asymptotically tends to the equilibrium state of the system as t tends to negative infinity. It is assumed that the kinetic energy, the Rayleigh dissipation function, and the positional forces in the neighborhood of the equilibrium position are infinitely differentiable functions. The results obtained here are partially generalized the results obtained by Kozlov et al. (Kozlov, V. V. The asymptotic motions of systems with dissipation. Journal of Applied Mathematics and Mechanics, 58^(5), 787–792 (1994). Merkin, D. R. Introduction to the Theory of the Stability of Motion (in Russian), Nauka, Moscow (1987). Thomson, W. and Tait, P. Treatise on Natural Philosophy, Part I, Cambridge University Press, Cambridge (1879)). The results are illustrated by an example.     

Penalty Methods for Numerical Approximations of Optimal Boundary Flow Control Problems

Abstract

This article is concerned with penalty methods for solving optimal Dirichlet control problems governed by the steady-state and time-dependent Navier-Stokes equations. We present, in two different versions, the penalized methods for solving the steady-slate Dirichlet control problems. These approaches are implemented and compared numerically. We also generalize the penalty methods to the time-dependent case. Scmidiscrete and fully discrete approximations of time-dependent Dirichlet control problems are discussed and implemented. Numerical results for solving both the steady-state and the time dependent Dirichlet control problems are reported.     

Director reorientation and order reconstruction: competing mechanisms in a nematic cell

We propose a model to explore the competition between two mechanisms possibly at work in a nematic liquid crystal confined within a flat cell with strong uniaxial planar conditions on the bounding plates and subject to an external field. To obtain an electric field perpendicular to the plates, a voltage is imposed across the cell; no further assumption is made on the electric potential within the cell, which is therefore calculated together with the nematic texture. The Landau-de Gennes theory of liquid crystals is used to derive the equilibrium nematic order tensor Q. When the voltage applied is low enough, the equilibrium texture is nearly homogeneous. Above a critical voltage, there exist two different possibilities for adjusting the order tensor to the applied field within the cell: plain director reorientation, i.e., the classical Freedericksz transition, and order reconstruction. The former mechanism entails the rotation of the eigenvectors of Q and can be described essentially by the orientation of the ordinary uniaxial nematic director, whilst the latter mechanism implies a significant variation of the eigenvalues of Q within the cell, virtually without any rotation of its eigenvectors, but with the intervention of a wealth of biaxial states. Either mechanism can actually occur, which yields different nematic textures, depending on material parameters, temperature, cell thickness and the applied potential. The equilibrium phase diagram illustrating the prevailing mechanism is constructed for a significant set of parameters.      

Application of a Mixed Variational Approach to Aeroelastic Stability Analysis of a Nonuniform Blade*

ABSTRACT

A mixed variational approach is used to investigate the coupled flap-lag-torsional aeroelastic stability of a nonuniform helicopter blade. This approach is particularly suited to treating rotating blades with sharp variations or discontinuous changes in their properties or cross-sectional dimensions. Numerical results by this approach are presented in hovering conditions, for both uniform and nonuniform blades with low torsional stiffness, and compared with the results by the usual Rayleigh-Ritz method. It is shown that the present approach has good convergence properties for both static equilibrium position and critical collective pitch angles.     

Prevalent Behavior of Strongly Order Preserving Semiflows

Classical results in the theory of monotone semiflows give sufficient conditions for the generic solution to converge toward an equilibrium or toward the set of equilibria (quasiconvergence). In this paper, we provide new formulations of these results in terms of the measure-theoretic notion of prevalence, developed in Christensen (Israel J. Math., 13, 255–260, 1972) and Hunt et al. (Bull. Am. Math. Soc., 27, 217–238, 1992). For monotone reaction–diffusion systems with Neumann boundary conditions on convex domains, we show the prevalence of the set of continuous initial conditions corresponding to solutions that converge to a spatially homogeneous equilibrium. We also extend a previous generic convergence result to allow its use on Sobolev spaces. Careful attention is given to the measurability of the various sets involved.     

Analytical Treatment of Some Extended Problems in Structural Optimization,Part I

ABSTRACT

Optimization problems involving supports of unspecified location as well as variable external actions and reactions of nonzero cost are discussed in the context of both plastic and elastic beams and frames, of non-preassigned, partially preassigned, and preassigned cost distribution, and of strength or deflection constraints.

The static-kinematic optimality criteria for various classes of problems are illustrated with simple examples and the results are checked by an independent method, i.e., by differentiation of the total cost with respect to the unknown variables. It is shown that for certain classes of problems the conditions introduced reduce to existing criteria by Prager and Masur.     

Nonequilibrium Effects During Spontaneous Imbibition

Accurate models of multiphase flow in porous media and predictions of oil recovery require a thorough understanding of the physics of fluid flow. Current simulators assume, generally, that local capillary equilibrium is reached instantaneously during any flow mode. Consequently, capillary pressure and relative permeability curves are functions solely of water saturation. In the case of imbibition, the assumption of instantaneous local capillary equilibrium allows the balance equations to be cast in the form of a self-similar, diffusion-like problem. Li et al. [J. Petrol. Sci. Eng. 39(3) (2003), 309–326] analyzed oil production data from spontaneous countercurrent imbibition experiments and inferred that they observed the self-similar behavior expected from the mathematical equations. Others (Barenblatt et al. [Soc. Petrol. Eng. J. 8(4) (2002), 409–416]; Silin and Patzek [Transport in Porous Media 54 (2004), 297–322]) assert that local equilibirum is not reached in porous media during spontaneous imbibition and nonequilibirium effects should be taken into account. Simulations and definitive experiments are conducted at core scale in this work to reveal unequivocally nonequilbirium effects. Experimental in-situ saturation data obtained with a computerized tomography scanner illustrate significant deviation from the numerical local-equilibrium based results. The data indicates: (i) capillary imbibition is an inherently nonequilibrium process and (ii) the traditional, multi-phase, reservoir simulation equations may not well represent the true physics of the process.     

A thermo-mechanical continuum theory with internal length for cohesionless granular materials

This article continues Part I. Here the non-equilibrium responses of the constitutive variables t (Cauchy stress tensor), q (heat flux vector), h (equilibrated stress vector), Γ (flux term associated with the internal length ℓ), Π (production term associated with ℓ) and f (equilibrated intrinsic body force) as well as the Helmholtz free energy Ψ are postulated by use of a quasi-linear theory for three of four models deduced in Part I. In so doing, together with the equilibrium responses gained in Part I, a complete set of constitutive equations for the constitutive quantities for each model is obtained. The implemented models are applied to investigate typical isothermal steady granular shearing flows with incompressible grains, namely, simple plane shear flow, inclined gravity-driven flow and vertical channel-flow. The emphasis is on the models in which ℓ is considered a material constant (Model I) and an independent dynamic field quantity (Model III). Numerical results show that Model III is more appropriate than Model I since in the former model the effect of the motion of an individual grain can better be taken into account. Such a result is in particular significant for avalanches, since it verifies the existence of a thin layer immediately above the base of an avalanche, in which the grains are colliding strongly with one another, and provides a quantitative means to measure such a thin layer.     

Overview of High-temperature Deformation Measurement Using Digital Image Correlation

Background

Developments in digital image correlation (DIC) in the last decade have made it a practical and effective optical technique for displacement and strain measurement at high temperatures.

Objective

This overview aims to review the research progress, summarize the experience and provide valuable references for the high-temperature deformation measurement using DIC.

Methods

We comprehensively summarize challenges and recent advances in high-temperature DIC techniques.

Results

Fundamental principles of high-temperature DIC and various approaches to generate thermal environment or apply thermal loading are briefly introduced first. Then, the three primary challenges presented in performing high-temperature DIC measurements, i.e., 1). image saturation caused by intensified thermal radiation of heated sample and surrounding heating elements, 2) image contrast reduction due to surface oxidation of the heated sample and speckle pattern debonding, and 3) image distortion due to heat haze between the sample and the heating source, and corresponding countermeasures (i.e., the suppression of thermal radiation, fabrication of high-temperature speckle pattern and mitigation of heat haze) are discussed in detail. Next, typical applications of high-temperature DIC at various spatial scales are briefly described. Finally, remaining unsolved problems and future goals in high-temperature deformation measurements using DIC are also provided. 

Conclusions

We expect this review can guide to build a suitable DIC system for kinematic field measurements at high temperatures and solve the challenging problems that may be encountered during real tests.

     

The general solution of one-dimensional hexagonal quasicrystal

A general solution is given for three-dimensional dynamic problems in 1D hexagonal quasicrystalline material with Laue class 6/m_hmm. In the general solution, the phonon and phason displacements and stresses are controlled by two displacement functions Ψ and F, which satisfy a second-order partial differential equation, and a sixth-order partial differential equation, respectively. The general solution for static equilibrium problems is treated as a special case.     

Boundary integral equations for inverse problems in the elasticity theory

The paper is concerned with the application of the boundary integral equation method for the holomorphic vector to nonclassical problems in the static theory of elasticity. Problems are considered for the case when on part of a body conditions are overvalued, i.e. the vectors of displacements and loads are prescribed, and on the other part of the body conditions are unknown (so-called (u, p) problem). It is shown that the method proposed is efficient and can be applied to the problems of computer defectoscopy in stationary potential fields.