Chapter 10 - Variational principles for nonlinear fluid–solid interactions

Abstract

The variational principles for linear fluid–solid interaction (FSI) problems were investigated. These principles provide a means for the transformation of the partial differential equations governing the dynamics of a structure, fluid, or FSI system, defined by an appropriate set of physical variables (i.e., displacement, pressure, stress) into an alternative set of ordinary differential or algebraic equations amenable to numerical analysis; hence this is a numerical scheme of study, as we have used effectively in one of the chapters to construct finite element models for linear FSI problems. However, these variational principles, based on linear assumptions, cannot be used to deal with nonlinear FSI problems. To understand this point, it is important to find the main differences in deriving variations of this functional for nonlinear FSI problems when compared with linear cases. For linear theories, we assume that the motions of the fluid and the solid are small, so that its original configuration is taken as our reference state, and there is no need to distinguish Lagrange and Euler coordinates, as well as the variations involved. For example, when we take the variation of a quantity defined in a fluid domain, we consider its boundary fixed at the original position and neglect the effect caused by boundary motion. Further, in the linear variation process, we always freely exchange the order of time and space integrations. For nonlinear cases, these are no longer valid assumptions because the large motions cause the boundary changes that have to be considered in the variation process. So that readers may learn these mathematical tools and derive the variational principles for nonlinear systems, the fundamental concepts of the variational process that are valid for nonlinear FSI systems will be discussed in detail in one of the subsections, following a short review of historical studies for the variations of nonlinear dynamic systems given in one of the subsections. Based on this knowledge and these methods, the variational principles, along with selected application examples for nonlinear FSI dynamics, are derived in the subsequent subsections. The generalized theory and mathematical works adopted here were presented by Xing and Price, and the applications to model the dynamics of nonlinear elastic ship–water interactions were given by Xing and Price. Recently, Zhou et al. investigated an interesting phenomenon on nonlinear low-frequency gravity waves in a water-filled cylindrical vessel subjected to high-frequency excitations using variational principles developed for nonlinear FSI systems.