Chapter 2. Formulation of Finite Element Method by Variational Principle

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1 Chapter 2 Formulation of Finite Element Method by Variational Principle

2 The Concept of Variation of FUNCTIONALS Variation Principle: Is to keep the DIFFERENCE between a REAL situation and an APPROXIMATE situation in MINIMUM REAL SITUATION APPROXIMATIE SITUATION Minimum In reality, we will find FUNCTIONALS to represent the Real and Approximate situations Functional function of functions Thus, derivation of appropriate FUNCTIONALS is an important step in FE formulations

3 Mathematical tool for determining MINIMUM of a function Minimum/maximum values of function determined by calculus: Given function: f(x) Procedures in determining maximum value of f(x): 1) Solve the equation df ( x) 0 dx with solution x x m The solution x m can be either maximum or minimum value of function f(x) 2) Check: If 2 d f dx ( x) 2 x x m > 0 X m is the minimum value of f(x) If 2 d f dx ( x) 2 x x m < 0 X m is the maximum value of f(x)

The Concept of Discretization The essence of FEM is Divide and Conquer - meaning if the geometry/loading/boundary conditions of entire medium is too complicated to be solved by existing tools, one

4 The Concept of Discretization The essence of FEM is Divide and Conquer - meaning if the geometry/loading/boundary conditions of entire medium is too complicated to be solved by existing tools, one viable way is to divide the continuum into a finite number of sub-divisions (elements) inter-connected at nodes. This process is called DISCRETIZATION Example of discretization : Estimate the land area of Antarctic: The land area of this continent is enclosed by complicated curved lines. One method for finding area is to use squares and rectangles enclosures over the entire area, because we know how to find the Enclosed areas of squares and rectangles. Antarctic By super impose the surveyed land mass of the Antarctic on square grids with each square mesh 40,000 km 2 counting Square meshes leading to total land area 16,180 million km 2 This land area of Anarchic obtained by the above method of discretization obviously is an approximated value. Because the actual area is 13.6% less than this approximated value.

5 Convergence of an Area Computation by Discretization There are two ways one may compute the approximation of areas by discretization: Method 1: Enclose the individual areas within the actual curved boundaries, and Method 2: Enclose the individual areas outside the actual curved boundaries. We observe that: The exterior envelope (Method 2) exceeds the actual values, and the interior envelope (Method 1) results in less than actual values. FE method often involves interior envelopes So, FE results are often less than the actual solutions.

NO available analytical solution Original

6 Application of Discretization Principle in FEM Example on WHY Discretization is necessary in real-world stress analysis: Manageable by analytical solution method NO available analytical solution Original geometry Discretized (approximate) geometry

7 Variational Principle in FEM MINIMUM for Close Approximation Original geometry Discretized (approximate) geometry The Difference in Results 0 but can be made MINIMUM VARIATIONAL PROCESS

Variational Process for General FE Formulation A continuum subject to ACTIONS with induced REACTIONS and boundary supports (conditions): Stress analysis Heat transfer Fluid dynamics ACTIONS: {P}: P

8 Variational Process for General FE Formulation A continuum subject to ACTIONS with induced REACTIONS and boundary supports (conditions): Stress analysis Heat transfer Fluid dynamics ACTIONS: {P}: P 1, P 2, P 3,, P n Forces {F}, pressures {p} Driving thermal forces: Q, q, etc. Induced REACTIONS: {Φ}: Φ 1, Φ 2, Φ 3,..,Φ m Actions and Reactions in Mechanical Engineering Analysis Local displacements {u}, Strains {ε}, stresses {σ} Local temperatures T Driving pressure Local velocities {V}

9 Variational Process for General FE Formulation-Cont d Real situation: Original geometry + loading/boundary conditions Difference expressed in FUNCTIONAL Discretized (approximate) geometry + loading/boundary conditions Variational Process for General FE Formulation: Minimizing the FUNCTIONAL Minimum

10 Functionals for Variational Process for General FE Formulation A continuum subject to ACTIONS with induced REACTIONS and boundary supports (conditions): Real Situation of solids Actions, Reactions and Functionals in Mechanical Engineering Analysis Stress analysis Heat transfer Fluid dynamics ACTIONS: {P}:P 1, P 2, P 3,,P n Forces {F}, pressures {p} Driving thermal forces: Q, q, etc. Induced REACTIONS: {Φ}: Φ 1, Φ 2, Φ 3,..,Φ m Local displacements {u}, Strains {ε}, stresses {σ} Approximate Situation with elements Local temperatures, T Driving pressure Local velocities V ( ) FUNCTIONAL Potential energy (P) Governing Diff. Equation plus boundary conditions Governing Diff. Equation plus boundary conditions

11 Real Situation on continuum Mathematical Modeling of Variational Process in Finite Element Analysis Formulation The FUNCTIONAL in the original continuum is: ( ) { } { } { } { } + s v s g dv f.,...,,..., r r Minimization of the functional will ensure the loaded continuum to be in equilibrium condition. Mathematically, This condition is satisfied by the relations: ( ) { } where v is volume, s is surface (boundary), r denotes x,y,z coordinates A number of equations for each induced reaction Φ: ( ) ( ) ( ) 0,... 0, 0, We will learn later that these are element equations for the discretized FE model

12 Mathematical Modeling of Variational Process in Finite Element Analysis Formulation In FE Model with finite number of ELEMENTS interconnected at NODES: Real Situation on solids The functional of the discretized continuum is: where e ( ) Approximate Situation on elements m e ( ) ( ) 1 Actions {P}: p 1, p 2, p 3, P n in discretized continuum Induced Reactions in elements: e e e e e { },,,..., : in ELEMENTS of discretized continuum is the functional in elements of discretized solid, and m is the total number of elements in the FE model The Variational process required for the equilibrium of the discretized continuum becomes: ( ) { } m 1 e( ) e { } from which the element equations for every element in the discretized FE model is derived 0 m

13 Element Equations in FE Model The element equations derived from the above Variational process usually have the form of: where ( ) ( ) ( ) 1 0, [ K e ]{ } { q} [K e ] coefficient matrix ( usually a square matrix) {Φ} matrix of unknown quantities at the nodes {q} the specified actions (or forces) at the nodes of the same element The unknown quantities at ALL nodes in the FE mesh can be obtained by assembling all element equations in the FE model, and result in the following OVERALL equation in the form: in which [ K ] [ ] 2 m 1 0, K e 3 0,... [ K ]{ } { Q} OVERALL coefficient matrix, and { Q} { q} n 1 with n total number of nodes in the FE model

14 SUMMARY OF VARIATIONAL PRINCIPLE Variational principle is used to minimize the difference in the approximate solutions obtained by the FE method on Discretized situation corresponding to the Real situations. Functionals are derived as the function to be minimized by the Variational process Functionals vary in the forms with the nature of the problems: functional for stress analysis of deformed solid structures is Potential energy, functional for heat conduction is the governing differential equation for heat conduction of solids functional for fluid dynamics is the differential equations called the Navier-Stoke s equation Outcome of the Variational process of discretized media is the element equations for each element in the FE model Element equations are assembled to form the OVERALL stiffness equations, from which one may solve for all Primary unknown quantities at all the nodes in the discretized media Therefore, it is not an over statement to refer the Variational principle to be the basis of FE method.

Finite Element Method in Geotechnical Engineering

Finite Element Method in Geotechnical Engineering Short Course on + Dynamics Boulder, Colorado January 5-8, 2004 Stein Sture Professor of Civil Engineering University of Colorado at Boulder Contents Steps