# 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 anal**

**ysis 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.**

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