The Wave Finite Element Method

ISBN-13: 9783540416388 / Angielski / Twarda / 2003 / 355 str.

B. F. Shorr; Galina V. Mel'nikova; F. B. Shorr

Opis

Computational mechanics, as a science employed for the numerical model ing of processes in nature and engineering, has over the last few decades developed two strands. The first concerns the putting of more and more powerful software packages into computational practice, using increas ingly high-performance computers with increasingly large memory. The traditional finite element and finite difference approaches are still preva lent. Over the years however, researchers have met with new problems; their solutions on the basis of traditional methods are at best difficult and at worst impossible to obtain. Such problems provided a powerful impetus in the development of the second strand, resulting in the development of es sentially new approaches for numerical modeling, for example meshless methods, "molecular" dynamics, neuron networks. The current state of the art formed the basis of many papers presented at the Fifth World Congress on Computational Mechanics, Vienna 2002. It is within the framework of the second strand that this book has been written."

Recenzje

From the reviews:

"This book is devoted to a new approach to the description of complex mechanical systems. ... among the actual nonclassical methods, the wave finite element method (WFEM) is of high interest. The author ... provides a first exposition of WFEM ... . The book is written very clearly, the graphs are excellent, and the reader can grasp the main facts from WFEM. The references are up-to-date ... . The book can serve as a reference text book for the students and researchers in mechanics." (Dumitru Stanomir, Zentralblatt MATH, Vol. 1063, 2005)

"I can say that the book contains a valuable collection of 1-D and some 2-D solutions for wave propagation in solids, and it made a valuable contribution towards a new general-purpose transient wave approach for solids. ... the book is truly recommended for everybody involved in problems that include explosions, shocks, seismic waves, and structures with suddenly varying properties at a time-scale close to the time a wave takes to propagate over the structure." (Technische Mechanik, Vol. 26 (2), 2006)

Spis treści

Theory.- 1 Foundation of the wave finite element method.- 1.1 Direct mathemetical modeling of wave propagation in an elastic rod.- 1.1.1 Background equations.- 1.1.2 Numerical examples.- 1.2 Wave approach to finite element modeling.- 1.2.1 Background equations of the wave finite element method (WFEM).- 1.2.2 Numerical examples.- 2 Simulation of simple one-dimensional wave processes.- 2.1 Longitudinal waves in a rod.- 2.1.1 Collision of rods of different sizes and mechanical parameters.- 2.1.2 Sudden stopping of a rod of a variable cross section.- 2.1.3 Wave propagation in a rod with inner elastic-inertial links.- 2.2 Torsional waves in a rod.- 2.2.1 Sudden stopping of a rotating shaft.- 2.2.2 Setting a disk in motion by sudden connection with a rotating shaft.- 2.3 Transverse waves in strings and cables.- 2.3.1 Waves in a string stretched by a constant force.- 2.3.2 Waves in a cable stretched by its own weight.- 3 Wave propagation in an inelastic rod.- 3.1 Longitudinal waves propagation in an inelastic rod.- 3.1.1 Discrete-continual model of an inelastic rod.- 3.1.2 Governing equations.- 3.2 Waves in a viscoelastic rod.- 3.2.1 Background equations.- 3.2.2 Numerical examples.- 3.3 Waves in an elastic-viscoplastic rod.- 3.3.1 Elastic-plastic models.- 3.3.2 An elastic-viscoplastic model.- 4 Coupled longitudinal-torsional waves in a pre-twisted rod.- 4.1 Basic equations.- 4.1.1 Governing equations for a pre-twisted rod.- 4.1.2 Wave model of a pre-twisted rod.- 4.2 Wave propagation induced by a force and torque.- 4.2.1 Waves induced by a constant load.- 4.2.2 Impulse-induced waves.- 5 Bending waves in a beam.- 5.1 Basic equations.- 5.1.1 Wave model of the Timoshenko beam.- 5.1.2 Finite element simulation of bending waves.- 5.2 Direct mathematical modeling of bending waves propagation.- 5.2.1 Structural bending/shear model of a beam.- 5.2.2 Solution procedure.- 5.3 Numerical examples.- 5.3.1 A stepped force affecting a beam.- 5.3.2 A stepped moment affecting a beam.- 5.3.3 Comparison of the DMM and WFEM approaches for bending waves modeling.- 6 One-dimensional waves in elastic continua and structures.- 6.1 Plane waves.- 6.1.1 Longitudinal waves.- 6.1.2 Transverse and coupled waves.- 6.2 Spherical and cylindrical waves.- 6.2.1 Spherical waves.- 6.2.2 Explosion in a spherical cavity of an elastic medium.- 6.2.3 Cylindrical waves.- 7 Numerical simulation of multi-dimensional wave processes.- 7.1 Foundation of the general WFEM approach.- 7.1.1 Governing equations.- 7.1.2 Waves in a plane region. Code WPRD.- 7 2 Numerical examnles.- 7.2.1 Sudden longitudinal loading of a one-side fixed plate.- 7.2.2 Sudden in-plane bending of a deep plate.- 7.2.3 A plate longitudinally impacted by a heavy body.- 7.2.4 A wide plate subjected to a bending moment.- 7.2.5 Additional remarks.- Applications.- 8 Impact loading of a deformable body.- 8.1 Principle of floating boundary conditions (FBC).- 8.1.1 Application of the FBC principle to WFEM.- 8.1.2 Special cases of body impact interaction.- 8.2 An elastic rod impacted by a rigid body.- 8.2.1 A rod of a constant cross section.- 8.2.2 The DMM accuracy in application to impact problems.- 8.2.3 A rod of variable cross section.- 8.3 An inelastic rod impacted by a rigid body.- 8.3.1 A rod of viscoelastic material.- 8.3.2 A rod of elastic-plastic material.- 8.4 Influence of contact deformation on impact response.- 8.4.1 Basic equations.- 8.4.2 Impact loading of a valve cylindrical spring.- 8.5 A pre-twisted rod impacted by a rigid body.- 8.5.1 Impact interaction of a rigid body with a pre-twisted rod.- 8.5.2 Lengthwise and turning impacts.- 9 Unsteady forced vibration of solids.- 9.1 Wave approach to study of forced vibration.- 9.1.1 Response of an elastic rod to harmonic excitation.- 9.1.2 Response of a rod of inelastic material.- 9.1.3 Transition through resonance domains under quasi-harmonic excitation.- 9.1.4 Response under fluctuating frequency and phase.- 9.2 Unsteady forced vibration of nonlinear systems.- 9.2.1 Torsional vibration of a shaft with a nonlinear clutch.- 9.2.2 Bending vibration of a turbine blade damped by a dry friction device.- 10 Unsteady vibro-impact loading.- 10.1 Multiple collisions at fixed points of a distributed system.- 10.1.1 Interaction of a rod with a viscoelastic foundation.- 10.1.2 Interaction of a rod with a hysteretic foundation.- 10.1.3 Switching on of a free-wheeling mechanism.- 10.2 Multiple collisions at varying points of a distributed system.- 10.2.1 Vibro-impact interaction of a string with limiters.- 10.2.2 A system with multiple inner gaps.- 11 Oscillations of a mechanical system affected by moving loads.- 11.1 General approach to simulation of moving loads.- 11.1.1 Equivalent node forces.- 11.1.2 Equivalent forces for different load/wave speeds ratio.- 11.2 Application of DMM to the study of 1-D waves induced by moving loads.- 11.2.1 A strip on a viscoelastic foundation.- 11.2.2 A beam on a viscoelastic foundation.- 11.3 Application of WFEM to the study of 2-D waves induced by moving loads.- 11.3.1 A long plate loaded by a transverse moving force.- 11.3.2 A long plate loaded by a longitudinal moving force.- 12 Dynamic loading of a free edge of a solid.- 12.1 Constant loads suddenly affecting a thin plate.- 12.1.1 A point force.- 12.1.2 A distributed load.- 12.2 Varying loads affecting a half-space.- 12.2.1 A point impulsive force.- 12.2.2 A distributed impulsive load applied to a limited domain.- 13 Some special problems of solid mechanics.- 13.1 Deformation of a chain of a varying length.- 13.1.1 Sliding down of an elastic chain under own weight.- 13.1.2 Numerical example.- 13.2 Waves in structures interacting with ‘active’ media.- 13.2.1 Strings on an ‘anti-elastic’ or ‘anti-viscous’ foundation.- 13.2.2 Auto-oscillation of a string in nonlinear viscous medium.- 13.2.3 Auto-oscillation in a system with intermittent contacts.- 14 Some special unsteady problems in engineering.- 14.1 Longitudinal dynamics of a train.- 14.1.1 Setting of a problem.- 14.1.2 Transient regimes of a train motion.- 14.2 Wave problems in adjacent areas of engineering.- 14.2.1 A transient process in an electrical circuit.- 14.2.2 Unsteady hydraulics problems.- Conclusion.- References.

Blurb

This monograph presents in detail the novel "wave" approach to finite element modeling of transient processes in solids. Strong discontinuities of stress, deformation, and velocity wave fronts as well as a finite magnitude of wave propagation speed over elements are considered. These phenomena, such as explosions, shocks, and seismic waves, involve problems with a time scale near the wave propagation time. Software packages for 1D and 2D problems yield significantly better results than classical FEA, so some FORTRAN programs with the necessary comments are given in the appendix. The book is written for researchers, lecturers, and advanced students interested in problems of numerical modeling of non-stationary dynamic processes in deformable bodies and continua, and also for engineers and researchers involved designing machines and structures, in which shock, vibro-impact, and other unsteady dynamics and waves processes play a significant role.