Some Theoretical And Computational Aspects Of Finite Element Method And Substructure-Subdomain Techanique For Dynamic Analysis Of The Coupled Fluid-Solid Interaction Problems

The principal results of investigations of this dissertation are:1. From the concept of four dimensional space, some general theorems for elastodynamics are obtained, for example, the principle of possible work action, the virtual stress-momentum principle, the virtual displacement principle, the theorem of strain-kinematic energy action, the reciprocal theorem for linear case and the displacement reciprocal theorem and the related theorems of time terminal conditions derived from it, etc. Under the four kinds of time terminal conditions, the variational principles of potential energy action and complementary energy action, the H-W principles, the H-R principles and the constitutive variational principles for elastodynamics are developed. The Hamilton's principle, Toupin's work and the formulations of the Ref. can be regarded as a special case of the general principles given in the dissertation. Using the action of momentum and its complementary one at time terminals, the variational principles for initial value problem of elastodynamics are presented and the initial conditions of problem are included in the Euler equations of the variational principles. By considering of three cases: piece-wise time domain, piece-wise space domain and piece-wise space domain and piece-wise space-time domain, the piece-wise variational principles including the potentional, the complementary, the mixed energy action forms are given. Finally, the general formulation of piece-wise variational principles is derived. If the time dimension of the four dimensional space is reduced, the results in the dissertation become the related variational principles for elaato-statics.2. In Comparison of the Ritz method with mode synthesis method and a finite element with a substructure, it is pointed that a substructure is essentially a "superelement" with Large number of internal freedom degrees. Regarding a substructure as a "superelement", and the boundaries of the substructure as the boundaries of the "superelement", applying and carring on further H.H. Pian's idea that had been used to formulate the models of finite element methods for elastostaties,we investigate the mechanical principle ofmode synthesis method, such as: formulation of substructure models, rule for mode reduction and error or convergence etc. Based on the variational principles for elasto-dynamies given in the chapter 1, there are four substructure models: the displacement consistence model the hybrid-dis-Placement model , the stress equilibrium model and the hybrid-stress model are formul- ated, of which, the last two models were given by author In order to illustrate the related analysis. a simple example and a numerical example are shown in the Chapter 2.3. By Comparing the time dimension with the space dimension of a four space and weighted residual method with variational method, the auther's general approach, which has been used to formulate time-element models by means of the generalized variational principles given in chapter 1 , is described in detail. This approach can be considered as a case to the Zienkiewiez's works but by using this approach, the time element, the space element, the four dimensional element and the related substructure models can be entirely formulated by the same approach - variational method. As an example, the generalized Newmark's formula and a single-step algorithm of time element are derived, and both of stability and accuracy of the two methods have been analysed.4. Applying the same idea of chapter 1 to the governing equations for linear theory of isotropic micropolar elasticity, such as Eringen's work , the related general theorems and variational, piece-variational principles under the various conditions at time terminals are drawn. As a special case of the principle of potential energy action, the variational formulation with the penalty coef-icient for restricting rotation vector of displacement field is naturally obtained. Physically, this penalty coeficient represents a rigidity modulus of micropolar rotation of materials. The influence of the penalty coeficient on the numerical results is discussed by the analysis and the numerical examples.5. Based upon the results in the chapter 1 and 4, the three basic variational formulations for dynamic analysis of coupled fluid-structure interaction problems are given. These are the functional of potential energy action with their field displacements in fluid and solid as variabls, the functional of complementary energy action with their stresses and momentums in fluid and solid as variabls and the functional of mixed energy action with the displacement in solid and the stress in fluid as variabls. According to these functionals, three finite element models: the displacement model, the equilibrium model and the mixed model for dynamic analysis of fluid-solid interaction problems, are formulated. For symmetrizing the non-symmetric finite element matrices obtained with the mixed model, the four methods have been proposed. More over, the related approximate formulations for analysis of a coupled dynamic problem are derived. These formulations are as follows: 1). the compressibility of fluid is neglected and its additional mass to structure is consideved;2) the mass of fluid is neglected and its additional rigidity to structure is considered. 3) the mass of structure is neglected and its additional flexibility to fluid is considered;4).the regidity of structure is neglected and its additional complementary mass to fluid is considered;In these four approximate methods, the first one is the common formulation. In a similar way to the chapter 2, the various models of substructure-subdomain for dynamic analysis of coupled problem are developed. Beslds the similar models to the Chapter 2, the mixed substructure-mibdomain model is also included in the chapter 6. It has been pointed that there are large numbers admissible mode sets for mixed substructure - subdomain model in accordance with the constrain conditions of the mixed functional principle. However, the two of these, the displacement consistent model and the pressure eauilibrium model, are mainly discussed. The table of all substructure-subdomain models for dvnamic analysis of coupled fluid-solid interaction problems is listed at the end of chapter 6.6. A computer program FSIAP consisting of about 7000 FORTRAN statements has been worked out. In the FSIAP. the finite displacement method and the displacement consistenev substructure- subdomain model has been adopted to calculated the modal parameters and the dynamic response of a solid, or a fluid, or a coupled fluid-solid system. Some of techniques, such as frequence shifting technique, "superelement" synthesis of substructure - subdomain analy-sis , penalty finite element method and a technique that is used to achieve the dispacement conditions at the fluid-solid interfaces, i.e., the normal dispacement at the interfaces mist be consisted and the tangent displacement are free, by means of the informations of freedom of the interface nodes, etc. are developed to overcome some difficulties in dynamic analysis of coupled fluid-solid problem. The subspace iteration method or the QL method can be used to calculate eigenvectors and mode superposition method or direct integration method can be used to calculate dynamic response. In a computational practice, users are able to choice one of the methods or the function described above by a correct writing the program cards. For the substructure - subdomain analysis function, the number of substructure or subdomain, their interface number and the interface position are not any additional constrains, and the synthesis computation will automatically implemented only if the related cards are correctly written.Although up to now there are only some plane elements in the FSIAP, there is no difficulty to develop it to some available programs because many related developable problems have been considered in the project of the FSIAP.Using the FSIAP, many examples are calculated, such as the pressure wave in fluid, the eigenvectors for some dam-water-base systems and their dynamic response to some movements of the dam base (using the EL CENTRO EARTHQUAKE DATA, MAY 18, 1940), etc. The results are listed in tables and figures and comparison of some of the results corresponding analytical results shows that the related methods might be considered as satisfactory.The dissertation contains 7 chapters, and about 150 References have been quoted.