A new implementation of the force method and the optimum design of large trusses

An implementation of the force method is proposed in which the forces and the displacements are simultaneously obtained by the solution of a sparse symmetric indefinite system. The matrix of coefficients is formed by just the concatenation of the element flexibility and equilibrium matrices. No computational procedure is required to generate the compatibility conditions (or the self-stress matrix) and no partitioning of the force vector is made into a basic set and a redundant set, unlike the conventional force method. A slightly modified sparse unsymmetric system can be written in which the stresses and the displacements are the unknowns. This is used as constraints in the formulation of the minimum weight design problem for large structures under static loading conditions. A sparse generalized reduced gradient package is used as the optimizer. A class of test problems involving large truss structures is solved. The results indicate that the present implementation of the force method is better than the displacement method for the optimum design of large structures.     

Form-finding of tensegrity structures with multiple states of self-stress

A numerical method is presented for form-finding of tensegrity structures with multiple states of self-stress. At the first stage, the range of feasible sets of the nodal coordinates and the force densities are iteratively calculated by the only known information of the topology and the types of members until the required rank deficiencies of the force density and equilibrium matrices are satisfied, respectively. The linear constraints on the force densities which are derived from the obtained configuration??s symmetry properties and/or directly assigned by designers are then utilized to define a single integral feasible force density vector in the second stage. An explanation on the null space of the force density matrix that generates the configurations of the tensegrities is rigorously given. Several numerical examples are presented to demonstrate the efficiency and robustness in searching new self-equilibrium stable configurations of tensegrity structures with multiple states of self-stress.     

Isotropic-BEM coupled with a local point interpolation method for the solution of 3D-anisotropic elasticity problems

This paper presents a solution procedure for the three-dimensional linear elastic problem with anisotropic properties. The approach uses the partition of the displacement field into complementary and particular parts. The former is the solution of a differential equation similar to that of an isotropic elastostatic and is obtained by the isotropic boundary element method. The particular integral is obtained by solving the corresponding strong form differential equations, using the local radial point interpolation method. This promising approach is simple to implement and leads to highly accurate solutions in some simple tested situations.     

Equivalence groups for second order balance equations

The groups of equivalence transformations for a family of second order balance equations involving arbitrary number of independent and dependent variables are investigated. Equivalence groups are much more general than symmetry groups in the sense that they map equations containing arbitrary functions or parameters onto equations of the same structure but with different functions or parameters. Our approach to attack this problem is based on exterior calculus. The analysis is reduced to determine isovector fields of an ideal of the exterior algebra over an appropriate differentiable manifold dictated by the structure of the differential equations. The isovector fields induce point transformations, which are none other than the desired equivalence transformations, via their orbits which leave that particular ideal invariant. The general scheme is applied to a one-dimensional nonlinear wave equation and hyperelasticity. It is shown that symmetry transformations can be deduced directly from equivalence transformations.     

Approximate solution of singular integro-differential equations in elastic contact problems

A method for the numerical solution of singular integro-differential equations is proposed. The approximate solution is sought in the form of the sum of a power series with unknown coefficients multiplied by a special term which controls the appropriate solution behaviour near and at the edges of the interval. The coefficients are to be determined from a system of linear algebraic equations. The method is applied to the solution of a contact problem of a disk inserted in an infinite elastic plane. Exact analytical solution is obtained for the particular case when the disk is of the same material as the plane. Comparison is made between the exact and the approximate solutions as well as with the solutions previously available in literature. The stability and the accuracy of the present method is investigated under variation of the parameters involved. The applicability of the method to the case when the boundary conditions for the unknown function are nonzero is discussed along with an illustrative example. A FORTRAN subroutine for the numerical solution of singular integro-differential equations is also provided.     

A Sign Control Method for Fitting and Interconverting Material Functions for Linearly Viscoelastic Solids

The multidata method was originally proposed to fit a Prony exponentialseries function to experimental viscoelastic modulus and compliance data;this was accomplished by the application of a linear least squares solver.This paper considers a similar approach, but extended in two key ways.First, it has been applied to the solution of convolution integralequations; specifically those used for material function interconversion.Second, it has been modified to force the signs of the Prony seriescoefficients to be positive; this is an essential criterion for the properphysical interpretation of a Prony series material function, which istypically not satisfied by multidata method solutions. Sign control isimplemented by an iterative Levenberg–Marquardt solution algorithm withan appropriate constraint, and can be used for both fitting experimentaldata and interconversion. To use the method, a Prony series must becapable of adequately representing the applicable functions. The method isdemonstrated by first fitting and converting experimental modulus data.Formulation for a composite lamina is also shown, in which a system ofintegral equations is reduced to a single integral equation; this can thenbe solved using the new method. Finally, application of the new method tofrequency domain transformations is demonstrated, along with comparisonsto other techniques.     

On certain classes of exact solutions of Einstein equations for rotating fields in conventional and nonconventional form

Using the symmetry reduction approach we have herein examined, under continuous groups of transformations, the invariance of Einstein exterior equations for stationary axisymmetric and rotating case, in conventional and nonconventional forms, that is a coupled system of nonlinear partial differential equations of second order. More specifically, the said technique yields the invariant transformation that reduces the given system of partial differential equations to a system of nonlinear ordinary differential equations (nlodes) which, in the case of conventional form, is reduced to a single nlode of second order. The first integral of the resulting nlode has been obtained via invariant-variational principle and Noether’s theorem and involves an integration constant. Depending upon the choice of the arbitrary constant two different forms of the exact solutions are indicated. The generalized forms of Weyl and Schwarzschild solutions for the case of no spin have also been deduced as particular cases. Investigation of nonconventional form of Einstein exterior equations has not only led to the recovery of solutions obtained through conventional form but it also yields physically important asymptotically flat solutions. In a particular case, a single third order nlode has been derived which evidently opens up the possibility of finding many further interesting solutions of the exterior field equations.     

A boundary element approach to some nonlinear equations from fluid mechanics

We consider a boundary integral approach to some nonlinear partial differential equations from fluid dynamics. The nonlinear equations are replaced by a sequence of linear equations, each of which is solved by the boundary element method. In order to avoid body integral contributions to the boundary integral equations, an approximate particular solution is first derived. This is achieved by replacing the body terms with an approximation for which there is a known solution. The present paper considers an approximation in terms of Gaussian distributions, a representation that has several desirable features.     

Mixed Convection of Non-Newtonian Erying Powell Fluid with TemperatureDependent Viscosity over a Vertically Stretched Surface

The viscosity of a substance or material is intensely influenced by the temperature, especially in the field of lubricant engineering where the changeable temperature is well executed. In this paper, the problem of temperature-dependent viscosity on mixed convection flow of Eyring Powell fluid was studied together with Newtonian heating thermal boundary condition. The flow was assumed to move over a vertical stretching sheet. The model of the problem, which is in partial differential equations, was first transformed to ordinary differential equations using appropriate transformations. This approach was considered to reduce the complexity of the equations. Then, the transformed equations were solved using the Keller box method under the finite difference scheme approach. The validation process of the results was performed, and it was found to be in an excellent agreement. The results on the present computation are shown in tabular form and also graphical illustration. The major finding was observed where the skin friction and Nusselt number were boosted in the strong viscosity.     

Analysis of 2D non-axisymmetric elasticity and thermoelasticity problems for radially inhomogeneous hollow cylinders

An analytical approach to solve plane static non-axisymmetric elasticity and thermoelasticity problems for radially inhomogeneous hollow cylinders is presented. This approach is based upon the direct integration method proposed by Vihak (Vigak). The essence of the method mentioned is in the integration of the original differential equilibrium equations, which are independent of the stress–strain relations. This gives the opportunity to deduce the relations, which are invariant with respect to various properties of the material, for the stress-tensor components. From these relations each of the stress-tensor components have been expressed in terms of the governing one. A solution of the equation for the governing stress in the form of Fourier series is presented. To determine the Fourier coefficients, an integral Volterra-type equation is derived and solved by a simple iteration method with rapid convergence. Other stress-tensor components are expressed through the obtained governing stress in the form of an explicit functional dependence on force and thermal loadings.     

Trefftz Methods for Time Dependent Partial Differential Equations

In this paper we present a mesh-free approach to numerically solving a class of second order time dependent partial differential equations which include equations of parabolic, hyperbolic and parabolic-hyperbolic types. For numerical purposes, a variety of transformations is used to convert these equations to standard reaction-diffusion and wave equation forms. To solve initial boundary value problems for these equations, the time dependence is removed by either the Laplace or the Laguerre transform or time differencing, which converts the problem into one of solving a sequence of boundary value problems for inhomogeneous modified Helmholtz equations. These boundary value problems are then solved by a combination of the method of particular solutions and Trefftz methods. To do this, a variety of techniques is proposed for numerically computing a particular solution for the inhomogeneous modified Helmholtz equation. Here, we focus on the Dual Reciprocity Method where the source term is approximated by radial basis functions, polynomial or trigonometric functions. Analytic particular solutions are presented for each of these approximations. The Trefftz method is then used to solve the resulting homogenous equation obtained after the approximate particular solution is subtracted off. Two types of Trefftz bases are considered, F-Trefftz bases based on the fundamental solution of the modified Helmholtz equation, and T-Trefftz bases based on separation of variables solutions. Various techniques for satisfying the boundary conditions are considered, and a discussion is given of techniques for mitigating the ill-conditioning of the resulting linear systems. Finally, some numerical results are presented illustrating the accuracy and efficacy of this methodology.     

Analysis of the equations of motion of linearized controlled structures

The linearized equations of motion of controlled structures possess coefficient matrices that lack the familiar properties of symmetry and definiteness. A method is developed for the efficient analysis of linearized controlled structures. This constructive method utilizes equivalence transformations in Lagrangian coordinates and does not require conversion of the equations of motion to first-order forms. Compared with the state-space approach, this method can offer substantial reduction in computational effort and ample physical insight. However, it is often necessary to draw upon some type of decoupling approximation for fast solution. Many numerical techniques involve discretized equations resembling those of linearized controlled structures. These numerical techniques can also be greatly streamlined if the method of equivalence transformations is incorporated. This research has been supported in part by the Alexander von Humboldt Foundation and by the National Science Foundation under Grant No. MSS-8657619. Opinions, findings, and conclusions expressed in this paper are those of the author and do not necessarily reflect the views of the sponsors.     

冗余并联结构六维力传感器及其超静定静力映射解析分析

提出采用冗余并联结构作为大量程大测力板六维力传感器的弹性体结构。基于整体刚度和变形协调条件,提出一种求解冗余n-SS(n>6)并联Stewart结构六维力传感器超静定受力分析的解析方法,导出了n个分支杆轴向力与六维外力之间的全映射关系。通过解析分析分块矩阵的广义逆,得到了当各分支杆刚度一致或不一致时形式统一的各分支杆轴向受力分析的求解方法。最后,利用数值算例验证了上述两种方法的正确性,并表明采用广义逆矩阵求解冗余并联结构各分支杆轴向力不仅形式简洁,而且易于计算机编程实现,具有重要的应用前景。     

Optimization in simulations of human movement planning

This paper discusses optimization algorithms in movement simulations for models of humans, humanoid robots or other mechanisms. Targeted movements between two configurations define a dynamically redundant system, for which there is freedom in the choice of control force time variations. A previously developed formulation for the treatment of targeted dynamics for mechanisms was used as a basis. The paper describes the development of an algorithm related to the method of moving asymptotes for the necessary optimization. The algorithm is specifically adapted to problems which are large and non‐linear but sparse, and which include very high numbers of design variables as well as constraints. In particular, non‐linear equality constraints from dynamic equilibrium equations are important. The optimization algorithm was developed to include these, but also in order to allow successively increasing penalty factors for constraint violations. The resulting setting was shown to be able to handle the systems established, robustly giving convergence to at least a local minimum also for very distant start iterates. The existence of very closely situated local optima, representing very similar movements, was discovered for the problem formulation, calling for an ad hoc method for finding the best of these local optima. Copyright © 2011 John Wiley & Sons, Ltd.     

SHAPE OPTIMIZATION OF A 2D BODY SUBJECTED TO SEVERAL LOADING CONDITIONS

An algorithm for shape optimization based on simultaneous solution of the equations and inequalities arising from Kuhn-Tucker necessary conditions is presented. Regular triangular FE assembly is proposed. Element vertices are associated with design variables directly or through spline parameters defining the boundary of the optimized body. This way, during the iteration procedure, FE assembly is automatically remeshed together with the motion of the optimized boundary. Multiple loading conditions are represented in the problem as equality conditions in the form of a set of equilibrium equations for each loading condition separately. From the necessary condition equations an additional, important relation between cost function, Lagrange multipliers associated with inequality constraints and their limit values is derived. The algorithm combines standard professional FEM programs with an optimizer proposed in the paper which is illustrated with shape optimization of several 2D bodies. The proposed approach is theoretically rigorous and relatively simple for practical applications, and allows considerations sensitivities, adjoint systems and constraints linearization to be avoided.     

Similarity transformations for the axisymmetric Boussinesque's problem

Summary Lie's infinitesimal transformation groups, which leave the basic equations of axially symmetric problems of classical elasticity invariant, are constructed. For the case of the axisymmetric Boussinesque's problem of an elastic semi-space subjected to a point force applied normal to its surface, the invariance of boundary and boundary conditions leads to the explicit form of similarity transformations which are used to solve the problem. Expressions for the displacements and stresses derived by this approach, which is believed to be new, are found to agree with the known results.     

A coupled BEM‐flexibility force method for bending analysis of internally supported plates

This paper presents a new method for the analysis of plates in bending with internal supports. The proposed method can be regarded as an extension of the well‐known force method (the flexibility matrix method) in the matrix analysis of structures. The solution is performed through two phases: the released plate phase, in which the plate is released from all internal supports and solved using the Boundary Element Method (BEM). The effect of internal supports is considered in the second phase, where a series of unit virtual loads is placed instead of the unknown redundant reactions at internal supports. The flexibility matrix is formed and compatibility of deformations at the locations of internal supports is satisfied. Hence, the corresponding system of equations is solved for the unknown redundant forces at internal supports. The final solution of the problem consists of the summation of two phases: the released plate phase and the cases of virtual unit loads phase. An efficient solution algorithm is developed to solve both phases simultaneously. The main advantages of the present formulation are: (1) the present formulation increases the versatility of the BEM as it allows the re‐usability of standard BEM codes for solution of plates in bending to be used in solving problems having internal supports, with even no modifications; and (2) the two solution phases are completely uncoupled; therefore it is easy to trace behaviour of the plate due to failure of one or more of the internal supports without re‐analysis. Several numerical examples are analysed. The results are compared to those of analytical and finite element models to demonstrate the accuracy and the validity of the present formulation. The present formulation is used also to study the differences between the finite element and boundary element modelling for building slabs. Copyright © 2002 John Wiley & Sons, Ltd.     

The embedded discontinuous Galerkin method: application to linear shell problems

A new discontinuous Galerkin method for elliptic problems which is capable of rendering the same set of unknowns in the final system of equations as for the continuous displacement‐based Galerkin method is presented. Those equations are obtained by the assembly of element matrices whose structure in particular cases is also identical to that of the continuous displacement approach. This makes the present formulation easily implementable within the existing commercial computer codes. The proposed approach is named the embedded discontinuous Galerkin method. It is applicable to any system of linear partial differential equations but it is presented here in the context of linear elasticity. An application of the method to linear shell problems is then outlined and numerical results are presented. Copyright © 2006 John Wiley & Sons, Ltd.     

On first integrals of second-order ordinary differential equations

Here we discuss first integrals of a particular representation associated with second-order ordinary differential equations. The linearization problem is a particular case of the equivalence problem together with a number of related problems such as defining a class of transformations, finding invariants of these transformations, obtaining the equivalence criteria, and constructing the transformation. The relationship between the integral form, the associated equations, equivalence transformations, and some examples are considered as part of the discussion illustrating some important aspects and properties.     

Mathematical formulation and numerical treatment based on transition frequency densities and quadrature methods for non-homogeneous semi-Markov processes

Non-homogeneous semi-Markov processes (NHSMP) are important stochastic tools for modeling reliability metrics over time for systems where the future behavior depends on the current and next states as well as on sojourn and process times. The classical method to solve the interval transition probabilities of NHSMPs consists of directly applying any general quadrature method to some non-convolution integral equations. However, this approach has a considerable computational effort. Namely, N²-coupled integral equations with two variables must be solved, where N is the number of states. Therefore, this article proposes a more efficient mathematical formulation and numerical treatment, which are based on transition frequency densities and general quadrature methods respectively, for NHSMPs. The approach consists of only solving N-coupled integral equations with one variable and N straightforward integrations. Two examples in the context of reliability are also presented. The first one addresses a case where a semi-analytical solution is available. Then an example of application concerning pressure-temperature optical monitoring systems for oil wells is discussed. In both cases, the proposed approach is validated via the comparison against the results obtained from the semi-analytical solution (for the first example) as well as from both the classic and the Monte Carlo methods.