# Notes on Numerical Modeling in Geomechanics

These “Notes” are the result of many years of experience in teaching, developing, and practicing numerical modeling in geomechanics; they are intended for beginners witha background in differential equations, mechanics of materials, matrix algebra, vector calculus, and some exposure to soil and rock mechanics.

The primary objective of these “Notes” is to develop a basic understanding of the finite element method (FEM) as used in geomechanics. FEM is well understood and has been in undergraduate engineering curricula for many years; it is by far the most popular numerical method for engineering design. Although the best way to learn FEM is by doing, that is, by writing a finite element computer program, learning a computer programming language such as Fortran or some version of C or another
high-level language is not the undergraduate engineering requirement that it once was.

Consequently, an approach based on development of FEM concepts and then use of available FEM programs to demonstrate applications is followed. A similar approach is followed in the case of the boundary element method (BEM) and the distinct element
method (DEM).

However, programming comments are given at the end of each chapter as additional guidance for those who are familiar with a high-level programming language such as Fortran. Otherwise, these comments are easily skipped in the progress of the “Notes”.
After a brief introduction in Chapter 1, an intuitive approach to the concept of a finite element is applied to interpolation over a triangle in Chapter 2. Derivatives of interpolation functions are discussed in Chapter 3. Linear interpolation over a quad-ilateral and derivatives are discussed in Chapters 4 and 5, respectively.

Element equilibrium and stiffness are developed in Chapter 6, followed by development of global equilibrium and stiffness in Chapter 7. Chapter 8 describes the concept of static condensation and the four-element constant-strain quadrilateral. Equation solving by elimination and iteration schemes is discussed in Chapter 9. Material nonlinearity and time integration are developed in Chapters 10 and 11. Fundamentals of seepage analysis and the finite element approach to hydroechanical coupling are described in Chapters 12 and 13, respectively. Basics of BEM are described in Chapter 14; DEM basics are described in Chapter 15.