On Graffi's reciprocal theorem in unbounded domains

A generalization of the reciprocal theorem of linear elastodynamics is proposed for a class of unbounded elastic bodies that may stiffen at infinity. It is achieved by means of some a-priori estimates derived in a previous paper of ours [1].     

A new uniqueness theorem of the linear thermo-elastic dynamics on unbounded domains

In the present paper, the uniqueness of the solution to the initial boundary value problem of the linear thermo-elastic dynamics on unbounded domains is obtained under less restrictive conditions, including abandoning the positive semi-definiteness of the elasticity tensor and boundness of the material tensor and restrictions on the acoustic tensor and the coupled tensor, and the results in [1] are refined. The conclusion here is valid for the case on bounded domains and the linear elastic dynamics on unbounded domains, hence the results in [2–4] are refined too. Abandoning the positive semi-definiteness of elasticity tensor permits that the uniqueness of the kinetic process is still valid for deformation of the wider materials, especially for the case that there are phase-transition during deformation process provided that the constitutive equations are unchanged in forms. The project is partially supported by The Youth Foundtion of Science of the Higher-Education of Shanghai and YFNSC (No. 19802012)     

Heat-flux-dependent thermoelasticity in unbounded domains

A work and energy equation and a reciprocity relation are derived for the heat-flux-dependent thermoelasticity theory of Lebon.     

On the work and energy theorem for unbounded elastic bodies

The main scope of the paper is the statement and the proof of the work and energy theorem for elastic bodies that occupy an unbounded region of space and whose acoustic tensor A may suitably grow at large spatial distance from a fixed origin.     

On calculation of hydrodynamic forces for steady flows in unbounded domains

This note addresses the question why the “impulse formula”, often employed to compute hydrodynamic forces in vortex-dominated time-dependent flows, is not applicable to steady flows in unbounded domains. By analyzing the asymptotic structure of steady and unsteady flow solutions in unbounded domains, it is demonstrated that one assumption made in the derivation of the impulse formula is in fact not satisfied in the steady case. This result also highlights the special character of steady flows in unbounded domains.     

Numerical modelling of unbounded domains in coupled fleld problems

Summary Two- and three-field problems are often defined in domains which may be assumed as unbounded. The traditional approach for their numerical simulation, within the framework of the finite element method, is by simple truncation of the mesh at a finite boundary. This fact both results in a large number of degrees of freedom and causes often errors in the analysis, due to the difficulty of setting correct conditions at the finite boundary.This paper shows the possible errors of the ensuing numerical solution and points out the usefulness of the infinite elements to simulate the far field response. Three examples from the field of isothermal and nonisothermal consolidation are presented where the improvements in the numerical simulation obtained by the use of infinite elements are evidenced. These examples may be considered as representative for a series of other coupled problems involving partial differential equations with first order time derivatives.

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Sommario Problemi di interazione fra due e tre campi sono spesso deflniti in domini che possono essere assunti come illimitati. Il modo tradizionale per la modellazione numerica di questi casi, nell'ambito del metodo degli elementi finiti, è quello di assumere frontiere fittizie in corrispondenza alle quali si devono imporre condizioni al contorno spesso di non facile valutazione. Questo modo di procedere comporta un numero di gradi di libertà assai elevato e può essere fonte di errori derivanti dall'imposizione di condizioni al contorno non corrette in corrispondenza della frontiera fittizia.Nel presente lavoro vengono evidenziati possibili errori delle conseguenti soluzioni numeriche e viene rimarcata l'utililità di elementi infiniti nella trattazione di questi problemi. In tre esempi di consolidazione isoterma e non, vengono messi in luce i miglioramenti della soluzione dovuti all'uso di elementi infiniti. Questi esempi sono rappresentativi di una più vasta classe di problemi accoppiati govemati da equazioni differenziali alle derivate parziali con derivate del primo ordine rispetto al tempo.

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Notation b body force vector - c coefficient of consolidation - c strain independent vector defining the creep strain rate - C _s specific heat of the solid phase - C _w specific heat of the fluid phase - D _T tangential stiffness matrix - g gravity acceleration - k absolute permeability matrix - k coefficient of thermal diffusivity - k _s bulk modulus of the solid phase - k _w bulk modulus of the fluid - L differential operator which relates displacements to strains - m (1 1 1 0 0 0) ^T - p pore pressure - Q _e volumetric outflow of the fluid per unit volume of the solid - Q _h outflow of heat per unit volume of solid - t time variable - dimensionless time parameter - T temperature increase over an equilibrium state - boundary traction vector - u displacement vector - V ^a apparent velocity of the fluid - z elevation above some datum - _s thermal expansion coefficient of the solid phase - _w thermal expansion coefficient of the fluid - total strain vector of the soil skeleton - ₀ represents all other strains not directly associated with stress changes - thermal conductivity matrix of the soil - dynamic viscosity - _s density of the solid - _w density of the fluid - effective stress in the soil skeleton - porosity Paper presented at the First Italian Meeting on Computational Mechanics (Milan, June 1986).     

SOLUTIONS FOR SECOND ORDER IMPULSIVE INTEGRO- DIFFERENTIAL EQUATION ON UNBOUNDED DOMAINS IN BANACH SPACES

Under loose conditions, the existence of solutions to initial value problem are studied for second order impulsive integro-differential equation with infinite moments of impulse effect on the positive half real axis in Banach spaces. By the use of recurrence method, Tonelii sequence and the locally convex topology, the new existence theorems are achieved, which improve the related results obtained by Guo Da-jun.     

Spatial decay estimates and upper bounds in elasticity for domains with unbounded cross-sections

In this paper we investigate spatial decay estimates and upper bounds for the solutions of elastic problems when the cross-sections of the three dimensional solid are semi-infinite strips. We obtain spatial decay estimates for the solutions of a static problem in the theories of homogeneous and isotropic linear elasticity and linearised elasticity. Energy bounds and some spatial decay estimates are obtained for the solutions of a dynamical problem in the case of anisotropic linear elasticity. For both kinds of problems we use the energy methods.     

Some remarks on continuous dependence and uniqueness in finite elastostatics for unbounded domains

Summary In connection with a foregoing paper, in the present note we establish some continuous dependence and uniqueness theorems for classical solutions to the equilibrium problem of finite elasticity. The elastic body is supposed to fill an unbounded domain of the physical space. We employ the weight function method and a suitable extension to the non-linear case of a method previously introduced by other authors for the linear case.

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Sommario In connessione con un precedente lavoro, si dimostrano nella presente nota alcuni teoremi di dipendenza continua ed unicità per soluzioni classiche del problema statico non lineare relativo ad un corpo elastico occupante un dominio illimitato dello spazio fisico. Vengono adoperati il metodo della funzione peso e l'estensione al caso non lineare di un metodo precedentemente introdotto da altri autori per il caso lineare.

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Work performed under the auspices of G.N.F.M. of the Italian Research Council (C.N.R.).     

通用有限元系统实现无限域波动仿真方法

在有限元模型中加入人工边界,可以进行无限域波动仿真。本文将介绍一种改进的Higdon人工边界,并通过适当的变换形式,将其在通用有限元软件上实现。最后通过数值实验验证了这种方法具有良好的稳定性和计算精度,并且实施简单方便,其他人工边界条件可参照本文方法,通过变换形式在通用有限元程序中实现。     

L2-decay for Navier-Stokes flows in unbounded domains,with application to exterior stationary flows

Communicated by C. Dafermos