A new family of time integration methods for heat conduction problems using numerical green’s functions

This paper is concerned with the formulation and numerical implementation of a new class of time integration schemes applied to linear heat conduction problems. The temperature field at any time level is calculated in terms of the numerical Green’s function matrix of the model problem by considering an analytical time integral equation. After spatial discretization by the finite element method, the Green’s function matrix which transfers solution from t to t + Δt is explicitly computed in nodal coordinates using efficient implicit and explicit Runge-Kutta methods. It is shown that the stability and the accuracy of the proposed method are highly improved when a sub-step procedure is used to calculate recursively the Green’s function matrix at the end of the first time step. As a result, with a suitable choice of the number of sub-steps, large time steps can be used without degenerating the numerical solution. Finally, the effectiveness of the present methodology is demonstrated by analyzing two numerical examples.     

A time-marching scheme based on implicit Green’s functions for elastodynamic analysis with the domain boundary element method

The present work presents an alternative time-marching technique for boundary element formulations based on static fundamental solutions. The domain boundary element method (D-BEM) is adopted and the time-domain Green’s matrices of the elastodynamic problem are considered in order to generate a recursive relationship to evaluate displacements and velocities at each time-step. Taking into account the Newmark method, the Green’s matrices of the problem are numerically and implicitly evaluated, establishing the Green–Newmark method. At the end of the work, numerical examples are presented, verifying the accuracy and potentialities of the new methodology.     

The analytic form of Green’s function in elliptic coordinates

The purpose of this study is the derivation of a closed-form formula for Green’s function in elliptic coordinates that could be used for achieving an analytic solution for the second-order diffraction problem by elliptical cylinders subjected to monochromatic incident waves. In fact, Green’s function represents the solution of the so-called locked wave component of the second-order velocity potential. The mathematical analysis starts with a proper analytic formulation of the second-order diffraction potential that results in the inhomogeneous Helmholtz equation. The associated boundary-value problem is treated by applying Green’s theorem to obtain a closed-form solution for Green’s function. Green’s function is initially expressed in polar coordinates while its final elliptic form is produced through the proper employment of addition theorems.     

Fundamental-solution-based hybrid FEM for plane elasticity with special elements

The present paper develops a new type of hybrid finite element model with regular and special fundamental solutions (also known as Green’s functions) as internal interpolation functions for analyzing plane elastic problems in structures weakened by circular holes. A variational functional used in the proposed model is first constructed, and then, the assumed intra-element displacement fields satisfying a priori the governing partial differential equations of the problem under consideration is constructed using a linear combination of fundamental solutions at a number of source points outside the element domain, as was done in the method of fundamental solutions. To ensure continuity of fields over inter-element boundaries, conventional shape functions are employed to construct the independent element frame displacement fields defined over the element boundary. The linkage of these two independent fields and the element stiffness equations in terms of nodal displacements are enforced by the minimization of the proposed variational functional. Special-purpose Green’s functions associated with circular holes are used to construct a special circular hole element to effectively handle stress concentration problems without complicated local mesh refinement or mesh regeneration around the hole. The practical efficiency of the proposed element model is assessed via several numerical examples.     

On the existence of periodic solution for equation of motion of thick beams having arbitrary cross section with tip mass under harmonic support motion

A cantilever beam having arbitrary cross section with a lumped mass attached to its free end while being excited harmonically at the base is fully investigated. The derived equation of vibrating motion is found to be a non-linear parametric ordinary differential equation, having no closed form solution for it. We have, therefore, established the sufficient conditions for the existence of periodic oscillatory behavior of the beam using Green’s function and employing Schauder’s fixed point theorem. The derived equation of vibration motion is found to be a non-linear parametric ordinary differential equation, having no closed form solution for it. To formulate a simple, physically correct dynamic model for stability and periodicity analysis, the general governing equations are truncated to only the first mode of vibration. Using Green’s function and Schauder’s fixed point theorem, the necessary and sufficient conditions for periodic oscillatory behavior of the beam are established. Consequently, the phase domain of periodicity and stability for various values of physical characteristics of the beam-mass system and harmonic base excitation are presented.     

Full-field analysis and image forces for piezoelectric bimaterials

Summary In this study, the two-dimensional explicit full-field solutions of transversely isotropic piezoelectric bimaterials subjected to mechanical and electrical loads are derived by using the Fourier-transform technique. The major objective of this study is to analyze the physical meaning and the structure of the solution. One of the novel features is that Green’s functions for bimaterials consist of Green’s functions for the infinite plane. The complete solutions of this problem include Green’s function of originally applied singularities in an infinite medium and nine image singularities which are induced to satisfy interface continuity conditions. It is shown that the physical meaning of the solution is the image method. The mathematical method used in this study provides an automatic determination for the locations of image singularities. The locations of image singularities are dependent on the roots of the characteristic equation for bimaterials. According to the characteristic roots, the number and distribution for image singularities are discussed in detail. The expressions for image forces acting on edge dislocations are given explicitly with the aid of the generalized Peach–Koehler formula. Numerical results for the full-field distributions of stresses, electric fields in bimaterials and image forces for edge dislocations are presented. Dedicated to Professor Franz Ziegler on the occasion of his 70th birthday     

A new technique to derive the Green’s type integral formula in thermoelasticity

New closed-form influence functions of a unit point heat source on elastic displacements and new Green’s type integral formula for a boundary-value problem (BVP) for a thermoelastic half-space are presented. The main difficulties in obtaining such results are observed in deriving the influence functions of a concentrated unit force onto elastic volume expansion and, also, in Green’s functions in heat conduction. For canonical Cartesian domains, these functions have been derived successfully for hundreds of BVPs and were published in a handbook. So, this paper shows the way to derive not only thermoelastic influence functions and Green’s type integral formulas for the half-space, but also for many new BVPs in thermoelasticity in other Cartesian canonical domains. Moreover, the technique proposed here may be applied in any orthogonal canonical domain provided by the lists of Green’s functions in heat conduction and influence functions for elastic volume expansion that are known.     

Nonideal contact problem of nonstationary heat conduction for two half-spaces

A nonideal contact problem of nonstationary heat conduction for two half-spaces with arbitrary initial conditions is considered. By introducing unknown conjugate functions and subsequent Laplace transformation, an integral representation for the problem solution has been found; equations in the region of transforms are solved with the aid of Green’s functions. In the particular case of exponential distributions of initial temperatures an accurate solution of the problem has been achieved. __________ Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 81, No. 2, pp. 373–383, March–April, 2008.     

A Study on Engineering History Base

During engineering processes, many kinds and amounts of information are used and produced. Such information is useful in successive cases, and thus its reuse is desired. In this paper, information on attributes of the products is referred to as ‘product information’, and information on the reasons for and/or history of a designer’s thought process is referred to as ‘process information’. According to this classification, process information can be used for obtaining an explanation of product information. The purpose of this work is to build a database, called an ‘engineering history base’, from which engineers can retrieve explanations to enable the reuse of product information. In this paper, explanation from the ‘process’ viewpoint is thought to be important in promoting the reuse of product information. There are two types of explanation for the process: teleological and causal. However, until now, little attention has been paid to the causal explanation for the process. Thus, in this paper, an information model which focuses on both types of explanation is proposed. The model is adopted for the engineering history base and a prototype system is developed. The appropriateness of this model is discussed by analyzing the actual data in the development of a color video printer.     

Hydroelastic Behavior of a Floating Ring-Shaped Plate

The interaction between incident surface water waves and floating elastic plate is studied. This paper considers the diffraction of plane incident waves on a floating flexible ring-shaped plate and its response to the incident waves. An analytic and numerical study of the hydroelastic behavior of the plate is presented. An integro-differential equation is derived for the problem and an algorithm of its numerical solution is proposed. The representation of the solution as a series of Hankel functions is the key ingredient of the approach. The problem is first formulated. The main integro-differential equation is derived on the basis of the Laplace equation and thin-plate theory. The free-surface elevation, plate deflection and Green’s function are expressed in polar coordinates as superpositions of Hankel and Bessel functions, respectively. These expressions are used in a further analysis of the integro-differential equation. The problem is solved for two cases of water depth infinite and finite. For the coefficients in the case of infinite depth a set of algebraic equations is obtained, yielding an approximate solution. Then a solution is obtained for the general and most interesting case of finite water depth analogously in the seventh section. The exact solution might be approximated by taking into account a finite number of the roots of the plate dispersion relation. Also, the influence of the plate’s motion on wave propagation in the open water field and within the gap of the ring is studied. Numerical results are presented for illustrative purposes.     

Theoretical Study of Interplay of Superconductivity and Ferromagnetism in Hybrid Rutheno–Cuprate Superconductors RuSr2RCu2O8

We consider the extended two-band s–f model with additional terms, describing intersite Cooper pairs’ interaction between 4f (5f) and conduction electrons. Following Green’s function technique and equation of motion method, self-consistent equations for superconducting order parameter (Δ) and magnetic order parameter (m _f ) are derived. The expressions for specific heat, density of states, and free energy are also derived. The theory has been applied to explain the coexistence of superconductivity and ferromagnetism in hybrid rutheno-cuprate superconductors RuSr₂RECu₂O₈ (RE = Gd, Eu). The theory shows that it is possible to become superconducting if the system is already ferromagnetic. A study of specific heat, density of states and free energy is also presented. The agreement between theory and experimental observations is quite satisfactory.      

Theoretical Study of Specific Heat,Density of States,Free Energy,and Critical Field of High Temperature Cuprate Superconductors Based on Intra and Interlayer Interactions

The role of attractive interlayer and intralayer interactions in layered high T _c cuprate superconductors have been investigated using a one-band two layer tight binding Hamiltonian. Self-consistent equations for the superconducting order parameter (Δ) and critical temperature (T _c ) are derived using double time Green’s functions and equation of motion method. The expression for excitonic type correlation (γ _c ), specific heat, density of states, free energy, and critical field are obtained. The interlayer interactions play an important role in the enhancement of T _c in layered high T _c cuprates. The oxygen isotope effect is also analyzed. The agreement between theoretical and experimental results for the system YBa_2−x La _x Cu₃O₇ (0≤x≤0.5) is quite satisfactory.      

Uniqueness and reciprocal theorems in linear micropolar electro-magnetic thermoelasticity with two relaxation times

A general model for the linear micropolar electro-magnetic thermoelastic continuum based on the hyperbolic heat equation, which is physically more relevant than the classical thermoelasticity theory in analyzing problems involving very short intervals of time and/or very high heat fluxes, is introduced. An integral identity that involves two admissible processes at different instants is established. Uniqueness theorem is proved, with no definiteness assumption on the elastic constitutive coefficients and no restrictions on the electro-elastic coupling moduli, magneto-elastic coupling moduli, and thermal coupling coefficients other than symmetry conditions. The reciprocity theorem is derived, without the use of Laplace transforms. The integral representation formula is obtained in case instantaneous concentrated, time-continuous or time-harmonic loads are applied. The Maysel’s, Somigliana’s and Green’s formulas are derived. The mixed boundary value problem is considered and a system of five singular Fredholm integral equations is obtained. The results for dynamic classical coupled theory can be easy deduced from the given general model formulated for the temperature-rate dependent thermoelasticity.     

Green’s functions for non-self-adjoint problems in heat conduction with steady motion

Heat conduction in a rectangular parallelepiped that is in steady motion relative to a fluid is studied in this paper. The governing equation consists of the standard heat equation plus lower-order derivative terms with the space variables that represent the effects of the solid flow. The presence of the first-order-derivative terms with the space variables renders the spatial part of the governing differenial equation non-self-adjoint and care must be exercised in defining the new Green’s functions to be used in representing the solutions of initial- and boundary-value problems. It is illustrated how the Green’s functions may be constructed and how solutions of initial- and boundary-value problems may be obtained that lead to numerical results. Convergence properties of the solutions are also discussed.     

Theoretical Study of Specific Heat, Density of States and Free Energy of Itinerant Ferromagnetic Superconductor URhGe

Following the equation of motion method and Green’s function technique, the coexistence of itinerant ferromagnetism (FM) and superconductivity (SC) is investigated in a single band homogeneous system. Self-consistent equations for superconducting order parameter (Δ) and magnetic order parameter (Δ_FM) are derived. It is shown that there generally exists a coexistent (Δ≠0 and Δ_FM≠0) solutions to the coupled equations of the order parameter in the temperature range 0<T<min (T _C,T _FM) where T _C and T _FM are respectively the superconducting and ferromagnetic transition temperatures. Expressions for the specific heat, density of states and free energy are derived. The specific heat has a linear temperature dependence at low temperatures as opposed to the exponential decrease in the BCS theory. The density of states for a finite Δ_FM increases as opposed to that of a standard ferromagnetic metal. The free energy shows that the superconducting ferromagnetic state has lower energy than the normal ferromagnetic state and therefore is realized at low enough temperature. The theory is applied to explain the observations of URhGe. The agreement between theory and experimental results is quite satisfactory.      

Coexistence of Itinerant Ferromagnetism and Superconductivity in ZrZn<Subscript>2</Subscript>

A microscopic coexistence of itinerant ferromagnetism and superconductivity is studied in a single- band homogeneous system, using an equation of motion method and Green’s function technique. Self-consistent equations for the superconducting order parameter Δ and magnetization m are derived. It is shown that there generally exists coexistent (Δ≠0 and m≠0) solutions to the coupled equations of the order parameters in the temperature range 0<T<min (T _c ,T _FM ), where T _c and T _FM are respectively the superconducting and ferromagnetic transition temperatures. The expressions for specific heat, density of states, free energy, and critical field are also derived. The theory is applied to explain the observations in ZrZn₂. The agreement between the theory and experimental observations is quite satisfactory.      

A simple orthotropic finite elasto–plasticity model based on generalized stress–strain measures

 In this paper we present a formulation of orthotropic elasto-plasticity at finite strains based on generalized stress–strain measures, which reduces for one special case to the so-called Green–Naghdi theory. The main goal is the representation of the governing constitutive equations within the invariant theory. Introducing additional argument tensors, the so-called structural tensors, the anisotropic constitutive equations, especially the free energy function, the yield criterion, the stress-response and the flow rule, are represented by scalar-valued and tensor-valued isotropic tensor functions. The proposed model is formulated in terms of generalized stress–strain measures in order to maintain the simple additive structure of the infinitesimal elasto-plasticity theory. The tensor generators for the stresses and moduli are derived in detail and some representative numerical examples are discussed. Received: 2 April 2002 / Accepted: 11 September 2002     

A Dominance-based Design Metric in Multiattribute Robust Design

This paper outlines the development of an effective and consistent ‘designing-in-quality’ strategy that can be used to deal with concepts of uncertainty, quality and robustness in engineering design. Specifically, this paper presents a decision analysis-based robust design metric that seamlessly integrates objective evaluations on the goodness of a design alternative with the designer’s intent and preferences. This is achieved through the development of a set of performance-reflecting dominance indices for the attributes and their utilization in a preference-influenced multiattribute utility formulation. Such a knowledge feedback-based decision model development will be particularly useful when dealing with complex iteration-based engineering design process where little information on the expected outcomes may be known a priori, or where product performance is computationally expensive to evaluate. Application of this robust design metric in a multi-stage experimentation and modeling design process is presented. The characteristics of the proposed design metric and the effectiveness of the overall design procedure in dealing with constrained engineering design problems are examined with the aid of demonstrative case studies and the results are discussed.     

The horn-feed problem: sound waves in a tube joined to a cone,and related problems

A semi-infinite tube is joined to a semi-infinite cone. Waves propagating in the tube towards the join are partly reflected and partly radiated into the cone. The problem is to determine these wave fields. Two modal expansions are used, one in the tube and one in the cone. However, their regions of convergence do not overlap: there is a region D{\mathcal{D}} near the join where neither expansion converges. It is shown that the expansions can be connected by judicious applications of Green’s theorem in D{\mathcal{D}}. The resulting equations are solved asymptotically, for long waves or for narrow cones. Related two-dimensional problems are also solved. Applications to acoustics, electromagnetics and hydrodynamics are considered.