ABSTRACT
Chapter 2 has shown various techniques to construct shape functions that can be used to assume functions for field variables such as the displacements in meshfree settings. Because of the diversity, functions assumed can have ‘‘unusual’’ behaviors that can ‘‘upset’’ usual functionals out of ‘‘control.’’ Therefore, directly using them in a standard weak formulation can be problematic. To ensure a meshfree method stable and convergent to the exact solution, the functions need to be first ‘‘managed’’ properly, adequate functionals need to be defined/constructed, and then proper formulation procedures have to be used accordingly. Types of weak forms will be discussed in Chapter 4. In this chapter we ‘‘manage’’ these functions by defining different spaces that host different types of functions. We start with the standard spaces that are widely used in usual weak formulations such as the finite element method (FEM) [1,2] and are familiar to many. We then provide a detailed discussion on G spaces that are the foundation of one weakened-weak formulation, are particularly useful for both meshfree and FEM settings, and much less familiar to many. Terminologies related to spaces are quite difficult for an engineer to comprehend, but for
a reasonably accurate and concise presentation of a numerical procedure, it is hard to avoid. The author, as an engineer, attempts to present these related terms in ‘‘engineering’’ languages for easier comprehension. The author has been trying hard to learn and study the formulations in both engineering and mathematics communities, and hope to do something to bridge the gap. If a reader can be patient in reading through this chapter, we hope he=she can find that the gap is getting a little smaller. In this chapter, we focus only on the minimum necessary terminologies, so as to facilitate
our discussions in later chapters without too much confusion and loss of accuracy in presentation. Readers may refer to Tables 1.1 and 1.3, when get lost in the mathematical terminologies and symbols.
