A numerical study of wave propagation in poroelastic media by use of the localized differential quadrature (LDQ) method

Founded 1905 A NUMERICAL STUDY OF WAVE PROPAGATION IN POROELASTIC MEDIA BY USE OF THE LOCALIZED DIFFERENTIAL QUADRATURE (LDQ) METHOD ZHANG JIAN (B. Eng., Dalian University of Technology, P. R. China) A THESIS SUBMITTED FOR THE DEGREE OF MASTER OF ENGINEERING DEPARTMENT OF MECHANICAL ENGINEERING NATIONAL UNIVERSITY OF SINGAPORE 2003 Acknowledgements I would like to thank both National University of Singapore and Institute of High Performance Computing for providing resources and funding for my research scholarship throughout this Master’s degree. There are many people to thank for their support and encouragement, without whom this thesis would not have been possible. Firstly, I would like to express my sincerest gratitude and appreciation to my supervisor Professor Lam Khin Yong for his dedicated support, invaluable guidance, and continuous encouragement in the duration of this thesis. His influence on me is far beyond this thesis and will benefit me a lot in my following life. I would also like to express my deep gratitude to my co-supervisor Dr. Zong Zhi for his enthusiastic help, kind consideration, great patience and invaluable guidance through the thesis. His dedication to research and vast knowledge inspire me in my future work. Next, there’re all the friends I’ve made and the list is too long to mention but their friendship, perspectives and recommendations helped me to make my stay in Singapore a really happy one. Special thanks to Zhang Yingyan, Yew Yong Kin, Si Weijian, Chen Jun, Wang Qingxia and Wang Zijie for their friendships and helps. Finally to my family, I appreciate their love, encouragement and support during my thesis. Without their strong and consistent support, it is impossible for me to finish this work. i Table of contents Acknowledgements i Table of contents ii Summary v Nomenclature vii List of Figures ix List of Tables xii Chapter Introduction 1.1 Background 1.2 Literature review 1.3 Objective and scope 1.4 Outline of the thesis 12 Chapter An introduction: poroelastic theory and the localized differential quadrature method 14 2.1 General 14 2.2 Poroelastic theory 15 2.2.1 Basic assumptions 15 2.2.2 The stress-strain relations in a fluid-saturated porous solid 16 2.2.3 Dynamic relations in the absence of dissipation 18 2.2.4 The governing equations of propagation of purely elastic waves 20 ii 2.3 DQ and its localizations in one- and two-dimension 20 2.3.1 DQ and its spatial discretization of the wave equation 21 2.3.2 Stability analysis 25 2.3.3 DQ localization in one dimension 28 2.3.4 DQ localization in two dimension 29 2.4 Comparison study of string vibration with DQ localization 30 2.5 Discretization of the governing equations 30 Chapter The numerical study of wave propagations in 1-D poroelastic media by use of the LDQ method 39 3.1 General 39 3.2 Wave propagation problems in 1-D poroelastic media 40 3.2.1 Vibration problem 40 3.2.2 Impact problem 42 3.2.3 Dynamic compatibility 45 3.3 Derivation of the analytical solutions of 1-D problems 46 3.4 Remarks 50 Chapter Wave propagations in 2-D and 2-D holed poroelastic media by use of the LDQ method 61 4.1 General 61 4.2 Wave propagation in 2-D poroelastic media 62 4.3 Wave propagation in 2-D holed poroelastic media 63 iii 4.3.1 The treatment of hole boundary conditions 63 4.3.2 Wave scattering over one square hole 65 4.3.3 Wave scattering over two square holes 67 4.3.4 Wave scattering over one circular hole 68 4.4 Remarks Chapter Conclusions 5.1 Concluding remarks 5.2 Recommendations for further research 69 98 98 100 References 102 List of publications 120 iv Summary The theory of elasticity describes the state of stresses and deformations in an elastic solid due to external forces or temperature changes. It has solved numerous problems, and provides a very powerful design and analysis tool in engineering. However, single use of elasticity theory cannot describe the mechanical behaviors of some materials such as cartilages and living bones which are made of an elastic matrix containing interconnected fluid-infiltrated pores. The right theory for such materials is poroelasticity. Poroelasticity is a continuum theory which models the interaction of solid deformation and fluid flow in a fluid-saturated poroelastic medium. Many investigations on poroelasticity have been conducted on static problems. But studies on wave propagation in poroelastic media are not enough yet. Motivated by this, the present thesis proposes the localized differential quadrature method to numerically simulate wave propagation in a poroelastic medium. First, the theory of propagation of elastic waves in fluid-saturated poroelastic media and the localized differential quadrature (LDQ) method are introduced. The poroelastic theory is briefly presented, and the governing equations of dynamic poroelasticity are simply presented in two-dimensional form. Differential Quadrature (DQ) method is a simple and highly efficient numerical technique which is characterized by approximating the derivatives of a function using a weighted sum of function values on a set of selected discrete points (grid points). However, DQ becomes more unstable if more grid points are used. To keep the balance between accuracy and stability, the localized differential quadrature (LDQ) method is proposed by applying DQ v approximation to a small neighbourhood of the grid point of interest rather than to the whole solution domain. Using this LDQ method, the discretization of the governing equations is solved together with fourth-order Runge-Kutta method. Second, wave propagations in one-dimensional poroelastic media subject to different loadings and support conditions are investigated by use of the LDQ method. The numerical results are compared with the closed-form analytical solutions derived using the technique of Laplace transform and inverse transform, where a very good agreement is obtained. Also, the parametric influence is investigated for the parameters of the coupling of volume change between the solid and fluid phases and the circular frequency to get a thorough understanding of the properties of wave scattering in poroelastic media. In accordance with Biot’s results, an interesting phenomenon, “dynamic compatibility”, is numerically shown to indirectly demonstrate the capability and validity of the method used herein. Finally, the studies on wave scattering in 2-D and 2-D holed poroelastic media are carried out. Since the analytical solutions to different problems subject to specified boundary and initial conditions are difficult to obtain, some comparative studies and analyses are made for wave propagation problems in holed media with different holes such as one square hole, two square holes and one circular hole. Numerical simulations effectively capture the characteristics of wave propagation. A linear interpolation method is presented to exemplify how to transform the boundary conditions on a circular hole to the neighbouring grid points in order to facilitate the numerical simulations. This method can be easily generalized to holes of other shapes. vi Nomenclature σij tensor of the stresses acting on the solid χ force component acting on the fluid p fluid pressure φ porosity εi j strain components in the solid ux , u y components of the solid displacements wx , w y components of the fluid displacements ε solid dilation e fluid dilation G potential energy per unit area of aggregate A Lamé coefficient N shear modulus R pressure related to volume change of the fluid into the aggregate of solid and fluid Q coupling between volume change of solid and fluid P volume change to normal stress in the solid K kinetic energy of the aggregate per unit area ρs mass density of the solid ρf mass density of the fluid vii ρa additional coupling density ρ11，ρ12，ρ 22 total effective mass of solid moving in the fluid and their coupling ρ total mass density of the fluid-solid aggregate qx total force on the solid per unit area in x -direction Qx total force on the fluid per unit area in x -direction u, w displacement vectors of solid and fluid phases aij , bij weighting coefficients for the first- and second-order derivatives (R ) , (R ) residuals for the first- and second-order derivatives t time T measured time ∆T time step ril , λ jl distances between two points in x - and y -direction S ij , Qij neighbourhoods of grid points in x - and y -direction v, η velocities of solid and fluid phases ω circular frequency h Laplace transform parameter (ˆ) Laplace transform E elasticity modulus µ Poisson’s ratio r n normal direction of hole edge viii List of Figures Figure 1.1 The schematic of a fluid-saturated porous bone structure: a spongy bone 13 Figure 2.1 DQ solution of the string vibration equation using 10, 15 and 18 grid points, respectively 35 Figure 2.2 Accuracy and stability relationship with the number of grid points, and the existence of a minimum on the curve of their sum 36 Figure 2.3 Localization of DQ approximation to a neighbourhood of the grid point of interest 37 Figure 2.4 String vibration simulated using localized DQ approximation. 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The schematic of a fluid-saturated porous bone structure: a spongy bone 13 Chapter 2 An introduction: poroelastic theory and the localized differential quadrature method Chapter 2 An introduction: poroelastic theory and the localized differential quadrature method 2.1 General The theory of propagation of elastic waves in fluid-saturated poroelastic media was originally established by Biot (195 6a) in. .. wave propagation in a column due to transient loading cases such as impact and step loadings Kumar and Hundal (2003) applied the method of characteristics to study the propagation of plane, cylindrical and spherical 6 Chapter 1 Introduction waves in a fluid-saturated incompressible porous medium Based on Biot’s theory, most of the investigations on wave propagations in poroelastic media have generally... facilitate the numerical simulations Numerical examples are given to analyse and compare the wave scattering characteristics 1.4 Outline of the thesis Chapter 2 presents the theory of propagation of elastic waves in fluid-saturated porous media and the proposed localized differential quadrature (LDQ) method in oneand two-dimension The technique to discretize the poroelastic governing equations using... yet Wave propagation in poroelastic media has wide applications A lot of our knowledge about the inner structure of the earth comes from seismic waves Careful analysis of the profiles of the seismic waves leads geography scientists to plot the density variations and compositions of the earth In medicine, by detecting the changes of the profiles of the waves in bone, doctors may locate the deep-seated... the shear modulus of the material The coefficient R is a measure of the pressure required on the fluid to force a certain volume of the fluid into the aggregate 17 Chapter 2 An introduction: poroelastic theory and the localized differential quadrature method while the total volume remains constant and Q is of the nature of a coupling between the volume change of the solid and that of the fluid Details... Rajapakse (1994) Rajapakse and Senjuntichai (1993) obtained the fundamental solutions for a poroelastic half-space under applied a static patch load and concentrated load using Laplace-Hankel integral transforms Zimmerman and Stern (1994) obtained several analytical solutions for some basic problems of harmonic wave propagation in a poroelastic medium Schanz and Cheng (2000) offered some solutions of. .. LDQ method to study wave propagations in 2-D and 2-D holed poroelastic media Linear interpolations are adopted to generalize the proposed method to 2-D poroelastic media with holes The phenomena of wave scattering over holes in fluid-saturated porous media are comparatively and qualitatively analysed Chapter 5 ends the thesis by concluding remarks and recommendations for further studies 12 Chapter 1 Introduction... elastic waves and in turn the sizes of the pores is small compared to the size of the element 4 The deformations are small enough so that linear elasticity theory (the small deformation theory) may be used to solid deformation 15 Chapter 2 An introduction: poroelastic theory and the localized differential quadrature method 5 The governing equations are derived in the absence of dissipation 2.2.2 The. .. observations From the stability analysis, we conclude that in spite of high 20 Chapter 2 An introduction: poroelastic theory and the localized differential quadrature method accuracy, DQ becomes more unstable if more grid points are used To keep balance between accuracy and stability, a localized differential quadrature (LDQ) method is proposed It is characterized by applying DQ approximation to a small... simulation of wave propagation always requires a high-precision scheme Based on such consideration, we propose a localized differential quadrature method to numerically simulate wave propagation in a poroelastic medium 1.2 Literature review The origin of the theory of poroelasticity can be traced back to the late eighteenth century when it was called porous media theory First contributions to the theory of . numerically simulate wave propagation in a poroelastic medium. First, the theory of propagation of elastic waves in fluid-saturated poroelastic media and the localized differential quadrature (LDQ). Chapter 4 Wave propagations in 2-D and 2-D holed poroelastic media by use of the LDQ method 61 4.1 General 61 4.2 Wave propagation in 2-D poroelastic media 62 4.3 Wave propagation in. Founded 1905 A NUMERICAL STUDY OF WAVE PROPAGATION IN POROELASTIC MEDIA BY USE OF THE LOCALIZED DIFFERENTIAL QUADRATURE (LDQ) METHOD ZHANG JIAN (B. Eng., Dalian University of Technology,