ngsolve::give me your element - portland state...

NGsolve::Give me your elementAnd let your eyes delight in my ways

Jay Gopalakrishnan

Portland State University

Winter 2015

Download code (works only on version 6.0) for these notes from here.

“Give me your heart / And let your eyes delight in my ways.” –The Bible

Jay Gopalakrishnan 1/18

http://web.pdx.edu/~gjay

http://web.pdx.edu/~gjay/teaching/mth610_2015/materials2.tar.gz

Contents

1 Automatic differentiation

2 Shape functions

3 Finite elements and spaces

4 Orientation

5 Bicubic qualdrilateral element


Automatic differentiation

Goal: Create variables that know how to (exactly) differentiate themselves.

Idea:

Differentiation obeys some rules (product rule, quotient rule etc) thatwe can implement by overloading operators, e.g., overload * toimplement

∂i (f ∗ g) = f (∂ig) + g(∂i f ).

Suppose an object representing xi (the ith coordinate) knows its valueand the value of its derivatives, at any given point. Then, we cancompute both the value and the value of derivatives of xi ∗ xj byoverloading * as above.


A minimalist class for differentiation

template<i n t D> c l a s s MyDiffVar // My d i f f e r e n t i a b l e v a r i a b l e s// F i l e : d i f f e r e n t i a b l e s . hpp

double Value ;double D e r i v a t i v e s [D ] ;

p u b l i c :

MyDiffVar ( ) ;

MyDiffVar ( double x i , i n t i ) // i−th c o o r d i n a t e x i hasValue = x i ; // grad = i−th u n i t v e c t o rf o r ( auto & d : D e r i v a t i v e s ) d = 0 . 0 ;D e r i v a t i v e s [ i ] = 1 . 0 ;

double GetValue ( ) const r e t u r n Value ;double& S e t V a l u e ( ) r e t u r n Value ;double G e t D e r i v a t i v e ( i n t i ) const r e t u r n D e r i v a t i v e s [ i ] ; double& S e t D e r i v a t i v e ( i n t i ) r e t u r n D e r i v a t i v e s [ i ] ;

;


Overload * for the differentiables

Template implementation of implement ∂i (f ∗ g) = f (∂ig) + g(∂i f ):

// implement p roduc t r u l e

template<i n t D> MyDiffVar<D>operator ∗ ( const MyDiffVar<D> & f , const MyDiffVar<D> & g )

MyDiffVar<D> f g ;

f g . S e t V a l u e ( ) = f . GetValue ( ) ∗ g . GetValue ( ) ;

f o r ( i n t i =0; i<D; i ++)f g . S e t D e r i v a t i v e ( i ) = f . GetValue ( ) ∗ g . G e t D e r i v a t i v e ( i )

+ g . GetValue ( ) ∗ f . G e t D e r i v a t i v e ( i ) ;r e t u r n f g ;

Quiz: Open the file and provide operators +,−, and /.


Using your class

#i n c l u d e ” d i f f e r e n t i a b l e s . hpp” // F i l e d0 . cppus ing namespace s t d ;

i n t main ( )

MyDiffVar<2> x ( 0 . 5 , 0 ) , y ( 2 . 0 , 1 ) ;

cout << ” x : ” << x << e n d l<< ” y : ” << y << e n d l<< ” x∗y : ” << x∗y << e n d l<< ” x∗y∗y+y : ” << x∗y∗y+y << e n d l ;

Using your simple class, you can now differentiate polynomial expressionsbuilt using x and y coordinates (or xi , i = 1, . . . ,N, in N-dimensions).


Exercise!

How would you modify differentiables.hpp so that you can alsodifferentiate expressions like sin(xy)?

Make sure your modified file compiles and runs correctly with this driver:

#i n c l u d e ” d i f f e r e n t i a b l e s . hpp” // F i l e d0x . cppus ing namespace s t d ;

i n t main ( )

MyDiffVar<2> x ( 0 . 5 , 0 ) , y ( 2 . 0 , 1 ) ;

cout << ” x : ” << x << e n d l<< ” y : ” << y << e n d l<< ” s i n ( x∗y )/ y : ”<< s i n ( x∗y )/ y << e n d l ;


Netgen’s AutoDiff class

An implementation of these ideas is available in$NGSRC/netgen/libsrc/general/autodiff.hpp.Here is an example showing how to use it:

#i n c l u d e <fem . hpp> // F i l e d1 . cppus ing namespace s t d ;

i n t main ( )

AutoDi f f<2> x ( 0 . 5 , 0 ) , y ( 2 . 0 , 1 ) ; // x and y coo rd sAutoDi f f<2> b [ 3 ] = x , y , 1−x−y ; // b a r y c e n t r i c coo rd s

cout << ” x : ” << x << e n d l<< ” y : ” << y << e n d l<< ” x∗y : ” << x∗y << e n d l<< ” x∗y∗y+y : ” << x∗y∗y+y << e n d l<< ” ( b0∗b1∗b2−1)/y : ” << ( b [ 0 ] ∗ b [ 1 ] ∗ b [ 2 ] − 1)/ y << e n d l ;

We will use AutoDiff and the following classes to program finite elements.


FlatVector, SliceVector, etc.

#i n c l u d e <b l a . hpp> // F i l e : f l a t v e c . cppus ing namespace s t d ; us ing namespace n g b l a ;

i n t main ( )

double mem [ ] = 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 1 0 ;

F l a t V e c t o r<double> f 1 ( 2 ,mem) ; // A v e c t o r c l a s s t ha t s t e a l sF l a t V e c t o r<double> f 2 ( 2 ,mem+3); // memory from e l s ewhe r e .cout << ” f 1 :\ n” << f 1 << e n d l ; // This p r i n t s 1 , 2 .cout << ” f 2 :\ n” << f 2 << e n d l ; // This p r i n t s 5 , 6 .

S l i c e V e c t o r<> s1 ( 4 , 2 ,mem ) ; // Al so s t e a l s memory .cout << ” s1 :\ n” << s1 << e n d l ; // This p r i n t s 1 , 3 , 5 , 7 .S l i c e V e c t o r<> s2 ( 5 , 1 ,mem+4); // What i s t h i s ?// :

SliceVector class does not allocate or delete memory.

Their constructors just create/copy pointers.


class ScalarFiniteElement

template < i n t D>c l a s s S c a l a r F i n i t e E l e m e n t : p u b l i c F i n i t e E l e m e n t

v i r t u a l vo id CalcShape ( const I n t e g r a t i o n P o i n t & ip ,S l i c e V e c t o r<> shape ) const = 0 ;

v i r t u a l vo id CalcDShape ( const I n t e g r a t i o n P o i n t & ip ,S l i c e M a t r i x<> dshape ) const = 0 ;

// . . . ;

shape and dshape are cheap to pass by value as function argumentseven when they contain many elements.

Any derived finite element class must provide shape functions andtheir derivatives.


Visualizing finite element “shape functions”

# FILE: shapes.pde

geometry = square.in2d

mesh = squareTrg.vol

fespace v -type=h1ho -order=2

gridfunction u -fespace=v

numproc shapetester nptest -gridfunction=u

The numproc shapetester is an NGsolve tool to visualize global basis functions(called global shape functions) of an FESpace.

Load this PDE file. Click Solve button before doing anything else.

Look for a tiny window called Shape Tester that pops up.

The number (0,1,. . . ) that you input in Shape Tester window determines whichbasis function will be set in gridfunction u.

Got to Visual menu and pick gridfunction u to visualize.


Visualizing finite element “shape functions”

# FILE: shapes.pde


mesh = squareTrg.vol

fespace v -type=h1ho -order=2

gridfunction u -fespace=v


1 , 2 , . . . 25 . . . etc.

Global shape functions


Prepare to write your own finite element

Study these files in the folder my_little_ngsolve:

myElement.hpp, myElement.cpp,myHOElement.hpp, myHOElement.cpp

All elements in a mesh are mapped from a fixed “reference element”.Pay particular attention to CalcShape(..) and CalcDshape(..).They give the values and derivatives of all local shape functions onthe reference element.

myFESpace.hpp, myFESpace.cpp,myHOFESpace.hpp, myHOFESpace.cpp

Each global degree of freedom (“dof”) gives a global basis functionand is associated to a geometrical object of the mesh (like a vertex,edge, or element). Pay particular attention to GetDofNrs(...),which return global dof-numbers connected to an element.


Homework

Your assignment is to code the bicubic finite element Q3 inbicubicelem.cpp. On the reference element, the unit square, thiselement consists of the space of functions

Q3 = spanx iy j : 0 ≤ i ≤ 3, 0 ≤ j ≤ 3.

Also code a bicubic finite element space (derived from FESpace), for anymesh of quadrilateral elements, in file bicubicspace.cpp

Then, use your space to approximate the operator −∆ + I and solve aNeumann boundary value problem. Tabulate errors.

The ensuing slides give you hints to complete this homework and suggestseparating the work into smaller separate tasks.


Bicubic shape functions on unit square

Task 1: In bicubicelem.cpp, provide shape functions.

E.g., here is a valid basis set of shape functions (you may use others) :

Ver

tex

bas

is φ0 = (1− x)(1− y)

φ1 = (1− x)y

φ2 = x(1− y)

φ3 = xy . . .

Ed

ge

bas

is

φ4 = (φ0φ1)φ0

φ5 = (φ0φ1)φ1

φ6 = (φ1φ3)φ3

φ7 = (φ1φ3)φ1

:. . .

Inte

rior

bas

is

φ4 = (1− x)(1− y)xyφ0

φ4 = (1− x)(1− y)xyφ1

: . . .Jay Gopalakrishnan 14/18

Bicubic shape functions on unit square

Task 1: In bicubicelem.cpp, provide shape functions.

Your basis expressions should go into the CalcShape memberfunction.

For the CalcDShape member function, you can use AutoDiff

variables and the same expressions you need in CalcShape.

Consider simplifying your code so that you only type the basisexpressions once.


Orientation

Task 2: In bicubicspace.cpp, write your finite element space.

Remember to keep track of matching local and global orientation (go back and revise

your bicubicelem.cpp if necessary).

/∗ What i s the l o c a l o r i e n t a t i o n ? I s the o r d e r i n g o f v e r t i c e s∗ and edges w i t h i n the r e f e r e n c e e l ement∗∗ v2 e1 v3 v3 e1 v2∗ o−−−−−−−−−o o−−−−−−−−−o∗ | | | |∗ e2 | | e3 e2 | | e3∗ | | | |∗ o−−−−−−−−−o o−−−−−−−−−o or someth ing∗ v0 e0 v1 , v0 e0 v1 , e l s e ? ∗/

What is the global orientation? NGsolve’s mesh edges

are directed/oriented. If edge shape functions from adjacent

elements are not given in that orientation, then you may lose

continuity!

↔↔


Check your basis

Task 3: Compile the code you wrote and make a shared library

make libmyquad.so

and check your basis functions on the three given quadrilateral mesh files.

# FILE : bicubicshapes.pde


#mesh = squareQuad1.vol.gz

#mesh = squareQuad2.vol.gz

mesh = squareQuad3.vol.gz

shared = libmyquad

define fespace v -type=myquadspace

define gridfunction u -fespace=v



Solve a PDE

Task 4: Using your finite element space, solve this boundary valueproblem:

−∆u + u = f on Ω

∂u/∂n = 0 on ∂Ω.

Hints:

Do you know the variational formulation for this problem?

You want to write a PDE file that mixes your finite element spacewith the NGSolve integrators.

E.g., the NGSolve integrator laplace, can work with any finiteelement which provides CalcDShape, by dynamic polymorphism.


Solve a PDE

Task 4: Using your finite element space, solve this boundary valueproblem:

−∆u + u = f on Ω

∂u/∂n = 0 on ∂Ω.

This task includes these steps:

1 Set f so that your exact solution is u = sin(πx)2 sin(πy)2.

2 Compute the L2(Ω) error (code this either in your own C++ numproc – like we didbefore – or find facilities to directly do it in the pde file).

3 Solve on mesh = squareQuad3.vol.gz by loading your pde file and pressing theSolve button. Compute the L2(Ω)-error. Note it down.

4 Pressing the Solve button again to solve and compute the L2-error on a uniformlyrefined mesh. Note the L2-error. Repeat (until you can’t).

5 What is the rate of convergence of L2-error with meshsize?


Project

Student Team Project: Learn about the “DPG method” and downloadan implementation of it in GitHub. Your job is to extend it to quadrilateralelements. You will need to code a new finite element space that will serveas the “test” space for the DPG method. Details will be progressivelymade clear as you proceed with the project.


https://github.com/jayggg/DPG

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