Numerical modeling of rock deformation: 06 FEM Introduction

Transcription

1 Numerical modeling of rock deformation: 6 FEM Introduction Stefan Schmalholz schmalholz@erdw.ethz.ch NO E 6 AS 9, Thursday -, NO D

2 Introduction We have learned that geodynamic processes can be described with partial differential equations (PDE s) derived from the concepts of continuum mechanics. These PDE s describing geodynamic processes are often complicated and non-linear and closed-form analytical solutions do not eist. Therefore, the PDE s are transformed into a system of linear equations which can be solved by a computer. The finite element method (FEM) is one method to transform systems of PDE s into systems of linear equations. Other methods are for eample the finite difference method, the finite volume method, the spectral method or the boundary element method. In this lecture the basic concept of the FEM is introduced by transforming a simple ordinary equation into a system of linear equations.

3 The model equation We consider the equation Case A) u ( ) A B u B L This equation is a second order, inhomogeneous ordinary differential equation. Physically it describes for eample: A) The deflection of a stretched wire under a lateral load (u = deflection, A = tension, B = lateral load) B) Steady state heat conduction with radiogenic heat production (u = temperature, A = thermal diffusivity, B = heat production) C) Viscous fluid flow between parallel plates (u = velocity, A = viscosity, B = pressure drop)

4 The analytical solution We find the general analytical solution by simple integration u ( ) B integrate d A u ( ) B C integrate d A B A u ( ) C C u The special solution is B BL A A u ( ) B L L=, A=, B=- The two integration constants are found by two boundary conditions u ( ) C BL u ( L) C A u

5 The weak formulation - The original equation u ( ) A B Multiply with test function u ( ) N( ) A B Integrate spatially L L u ( ) N( ) A B d u ( ) N( ) A N( ) B d Integrate by parts L L u ( ) N ( ) u ( ) N( ) A A N( ) B d u ( ) LN ( ) u ( ) N( ) A d A N( ) Bd L u ( ) L N ( ) u ( ) N( ) A A N( ) Bd The product rule of differentiation a b b a ab b a a ab b u ( ) a N( ), b A

6 The weak formulation - L u ( ) L N ( ) u ( ) N( ) A A N( ) Bd The first term requires our function u() at the boundaries = and =L and is given by the boundary conditions. Therefore, we do not need to test u() at these boundary points. L N( ) u( ) A N( ) Bd The original equation u ( ) A B The above equation is the weak form of our original equations, because it has weaker constraints on the differentiability on our solution (only first derivative compared to second derivative in original equation).

7 The finite element approimation - We approimate our unknown solution as a sum of known functions, the so-called shape functions, multiplied with unknown coefficients. The shape functions are one at only one nodal point and zero at all other nodal points, so that the coefficient corresponding to a certain shape function is the solution at the node. u ( ) u i ui i i i i

8 The finite element approimation - We approimate our unknown solution as a sum of known functions, the so-called shape functions, multiplied with unknown coefficients. The shape functions are one at only one nodal point and zero at all other nodal points, so that the coefficient corresponding to a certain shape function is the solution at the node. ui u( ) uin( ) i ui N( ) i N( ) i N( ) i ui T Nu u ( ) u i ui i i i i

9 The finite element approimation - We approimate our unknown solution as a sum of known functions, the so-called shape functions, multiplied with unknown coefficients. The shape functions are one at only one nodal point and zero at all other nodal points, so that the coefficient corresponding to a certain shape function is the solution at the node. ui u( ) uin( ) i ui N( ) i N( ) i N( ) i ui T Nu u ( ) i N( ) i d i i N( ) i i d i i u i ui i i Finite Element d i i i i

10 The finite element approimation - LN( ) u( ) A N( ) Bd d N( ) i ui N( ) i A N( ) ( ) i N i Bd N( ) i ui N( ) i u N( ) i N( ) i N( ) i d N( ) ( ) u ( ) i N i i N i A Bd N( ) i ui N( ) i N( ) i N( ) i N( ) i N( ) i d ui N( ) i A Bd N( ) i N( ) i N( ) i N( ) i ui N( ) i N( ) i N( ) i N( ) i N( ) i d ui d N( ) i Ad Bd N ( ) i N ( ) i N ( ) i N ( ) i u i N( ) i Ku - F u i ui The FEM where the shape functions are identical to the test (weight) functions is called Galerkin method after Boris Galerkin.

11 The finite element approimation - Ku - F K N( ) i N( ) i N( ) i N( ) i N( ) Bd d d i Ad, F N( ) i N( ) i N( ) i N( ) i N( ) i d N( ) ( ) d i N i Kii Ad A d d d d A A A d d d d i N( ) i d A A Bd d d, K F A A Bd d d i N( ) i N( ) i d i N( ) i i i i i d d i

12 The finite element approimation - 4 Local system for element A A Bd d d u A A Bd u d d u Local system for element u A A Bd u d d u A A Bd d d d L/ d L Global system for sum of all elements A A Bd d d u A A A ubd d d d u Bd A A d d

13 The finite element approimation - 5 Global system without boundary conditions A A Bd d d u A A A u Bd d d d u Bd A A d d d L/ d L Implementing boundary conditions: u and u = u A A A u Bd d d d u

14 The finite element approimation - 5 Global system without boundary conditions A A Bd d d u A A A u Bd d d d u Bd A A d d Implementing boundary conditions: u and u = u A A A u Bd d d d u d L/ d L Calculate solution using L= (d=l/=5), A=, B=- u u u u 5 u 5.5 u B BL A A u ( ) Analytical solution at =5 u ( 5) 5 5.5

15 Programming finite elements - Global nodes Local nodes Nine finite elements Which nodes belong to what element? Introduce a node matri! The NODE matri assigns to each local element the corresponding global node Ten nodes i i D ecursion

16 Programming finite elements - Nine finite elements Global nodes Ten nodes Local nodes Local system for 4 finite element d d A A Bd d d d d u A Au Bd d d A A B d u4 A A u 5 Bd

17 Programming finite elements - The global stiffness matri is the sum of all local stiffness matries. In our code we will first setup a global stiffness matri K Global full of zeros. We will then ) loop through the finite elements, ) identify where the local element is positioned within the global matri and ) add the local stiffness matri at the correct position to the global matri. A A d d A A A A d d A A d d A A d d d d A A d d K Global Eactly the same is done for the right hand side vector F.

18 The FEM Maple code > restart; > # FEM # Derivation of small matri for D nd order steady state > # Number of nodes per element nonel := : > # Shape functions N[] := (d-)/d: N[] := /d: > # FEM approimation u := sum( t[j]*n[j],j=..nonel): > # Weak formulation of D equation for i from to nonel do LN( ) u( ) A Eq_weak[i] := int( -A*diff(u,)*diff(N[i],) + B*N[i],=..d); od: > # Create element matries for conductive and transient terms K:=matri(nonel,nonel,): F:=matri(nonel,,): for i from to nonel do for j from to nonel do K[i,j] := coeff(eq_weak[i],t[j]): od: F[i,] := -coeff(eq_weak[i],b)*b: od: > # Display K and F matri(k); matri(f); A d A d A d A d N( ) Bd B d Bd

19 The FEM Matlab code In the numerical solution all parameters can be made variable easily, while it gets more difficult to find analytical solutions for variable parameters, e.g., A and B.

20 FEM applications Taken from MIT lecture, de Weck and Kim

21 FEM element types Taken from unknown web script

22 FEM applications Shortening of the continental lithosphere with: (i) viscoelastoplastic rheology, (ii) transient temperature field, (iii) viscous heating and (iv) gravity.

23 FEM applications Three-dimensional viscous folding