A PSE for Automatic Matlab 3DFinite Element Code Generation and Simplified Grid Computing

Zhou Jun, and Yukio Umetani

Shizuoka University of Japan

Outline

• Motivation

• Previous work

• Automatic Matlab 3D FEM code generation

• Execution of Matlab script via UNICORE

• Conclusions and future work

Motivation

Interface to Matlab

• popular for matrix computation• visualization facility• verifying simulation results of PSILAB

Reduce Matlab FEM programming effort• coding time• learning time• programming errors

Run Matlab script on the Grid via UNICORE

GridPSi overview

GridPSi client connects GridPSi server in three ways.

Remote Server

GridPSi Server

GridPSi Server

Unicore Client

GridPSi Client

Local Machine

Usite

Unicore Gateway

NJS

TSI

GridPSi Server

Internet

Internet

PSILABPSILAB

MATLAB

PDE wizard-based problem modeling

GridPSi client arcitecture

PSILAB Script Generator

MATLAB Script Generator

& Self-designed Matlab

Functions

Automatic generation of Matlab 3D FEM code

• MATLAB Script Generator

• Self-coded Matlab functions

Matlab Script Generator

Self-coded Matlab functions Generic

Data structures

PSILAB Script Generator

Example problem

To investigate the vertical movements at the bridge area of a 3D piano soundboard of simplified shape that is subjected to vertical pressures caused by the vibrations of the strings. (The soundboard is in anisotropic material)

Mathematical model

• Governing equations: MÜ + CÙ + KU = R (1)

]12020[)2cos(100

1 22 yxfqtsz (2)

• Fixed Dirichlet conditions on the four sides• Generalized Neumann conditions on the top and bottom

• External forces:

• Boundary conditions:

• Initial conditions:

No displacements at the initial state

Finite element solution

• Solid linear tetrahedron elements

• Central difference method

UCt

Mt

UMt

KRUCt

Mt

tttttt

2

112

2

11222

Matlab FEM data structures

Nodal coordinate matrix

Element connectivity matrices • Problem domain

• Boundary regions

stiffness, mass, and damping matrices

Nodal force vector

Surface tractions (Neumann boundary condition)

Unknown field on boundaries (Dirichlet condition)

Build nodal coordinate and element connectivity matrices

function [node, setS, setV] = gmsh2matlab(‘meshFile’)

• node: a matrix of nodal coordinates (n-by-3)

• setS: a set of element connectivity matrices of boundary surfaces

• setV: a set of element connectivity matrices of the volume(s)

• meshFile: The input mesh file

Build stiffness matrix

function [K] = buildStiffnessMatrix3D(node, elemV, E)

kTkkk

Gn

k

e EBBJK

1

(3)

E: the consistent stress-strain compliance matrixfunction [E] = buildComplianceMatriceElastic3D(material properties)

nG, Wk : integration point and weight

function [W,Q] = quadrature(order,type,dimension)

Jk: the determinant of the Jacobian matrix

Bk : the bar matrix of the element

function [N,dNdxi]=lagrange_basis('H4',pt) %return the Lagrange interpolant basis

%and its gradients

4321 BBBBBk

xizi

yizi

xiyi

zi

yi

xi

i

NN

NN

NN

N

N

N

B

,,

,,

,,

,

,

,

0

0

0

00

00

00

function [K] = buildStiffnessMatrix3D(node, elemV, E) n = size(node,1); % get the total node number ‘n’. m = size (ElemV,1); % get the total element number ‘m’.K =sparse (3*n,3*n); % create the sparse global stiffness matrix ‘K’.for e = (1:m) % loops over the elements. str = ElemV(e,:); % get the four nodes comprising the current element. % the following is to build the Bar array ‘strBK’, which is used to locate the positions % of the four nodes in the global stiffness matrix. strBk = [str(1) str(1)+n str(1)+2*n str(2) str(2)+n str(2)+2*n ... str(3) str(3)+n str(3)+2*n str(4) str(4)+n str(4)+2*n]; [W,Q] = quadrature(1,'TRIANGULAR',3); % A function call returns the quadratrue weight W and Q for q = 1:size(W,1) pt = Q(q,:); wk = W(q); [N,dNdxi]=lagrange_basis('H4',pt); % A function call returns the lagrange interpolant basis. Jacobian = node(str,:)'*dNdxi;% get the Jacobian matrix. dNdX = dNdxi * inv(Jacobian); % a 4-by-3 matrix of the shape functions. %% construct the stress-displacement matrix BK (a 6-by-12 matrix) %% Bk = [dNdX(1,1) 0 0 dNdX(2,1) 0 0 dNdX(3,1) 0 0 dNdX(4,1) 0 0; 0 dNdX(1,2) 0 0 dNdX(2,2) 0 0 dNdX(3,2) 0 0 dNdX(4,2) 0; 0 0 dNdX(1,3) 0 0 dNdX(2,3) 0 0 dNdX(3,3) 0 0 dNdX(4,3); dNdX(1,2) dNdX(1,1) 0 dNdX(2,2) dNdX(2,1) 0 dNdX(3,2) dNdX(3,1) 0 dNdX(4,2) dNdX(4,1) 0; 0 dNdX(1,3) dNdX(1,2) 0 dNdX(2,3) dNdX(2,2) 0 dNdX(3,3) dNdX(3,2) 0 dNdX(4,3) dNdX(4,2); dNdX(1,3) 0 dNdX(1,1) dNdX(2,3) 0 dNdX(2,1) dNdX(3,3) 0 dNdX(3,1) dNdX(4,3) 0 dNdX(4,1)];

K(strBk,strBk) = K(strBk,strBk) + Bk’ *E * Bk * wk * det(Jacobian); % assemble the global matrix. endend

Build mass and damping matrices

function [M] = buildMassMatrix3D(node, elemV ,rho)

NdVNM TV

eC

4

4

4

3

3

3

2

2

2

1

1

1

0

0

0

0

0

00

0

0

0

0

00

0

0

0

0

00

0

0

0

0

0

N

.

,

20

10jiif

jiifV

V

dVeV ji

(the integral over the element)

(4)

Enforce boundary conditions

function [v] = applyEssentialBCs(v,node,elemS,id,dx,dy,dz)

• Dirichlet boundary conditions:

• Neumann boundary conditions:

function [f]=calculateSurfaceTraction(node,elemS,id,px,py,pz)

u t+1 = u t + 2Δt*v; % update displacement vector

R = f body + f surface traction1 + f surface traction2 + …

)(t2

1 v UU ttttt

(5)

Initial Conditions

function [Uold,Unow]= initalizeDisVectors(Uold,Unow,initU,initV)

• Uold : the displacement vector at time t-1

• Unow : the displacement vector at time t

• initU : the initial value at time -1

• initV : the initial value at time 0

Solution Schemewhile (CT <500) t = t+delt; CT = CT+1; % Apply the Neumann conditions on given surfaces. [f1] = caculateSurfaceTraction2(node,elemS30,'30',0,0,-sz); [f2] = caculateSurfaceTraction2(node,elemS8,'8',0,0,0); R = f + f1 + f2; Unew = (a0*M+a1*C)\(Feffective - (K-a2*M)*Unow - (a0*M-a1*C)*Uold); Vnow = a1 * (Unew - Uold); % Apply the Dirichlet conditions on given surfaces. [Vnow] = applyEssentialBCs(Vnow,node,elemS17,'17',0,0,0); [Vnow] = applyEssentialBCs(Vnow,node,elemS21,'21',0,0,0); [Vnow] = applyEssentialBCs(Vnow,node,elemS25,'25',0,0,0); [Vnow] = applyEssentialBCs(Vnow,node,elemS29,'29',0,0,0); Unew = a3 * Vnow +Uold; Uold = Unow; Unow = Unew; Uzz = Unew([2*n+1:3*n],1);% get the displacements in Z direction. end % fid =fopen('matlabResults.txt','w'); fwrite(fid,Uzz); fclose(fid);

Executing Matlab script via UNICORE

• Use of Script Task plug-in interface of UNIOCRE client

• Preparation of a Job containing two tasks with

dependency

1. script task:

• command: matlab -nosplash -nodesktop -nodisplay -r soundboard

• import files: the generated script, predefined Matlab functions, the mesh file

2. Export task: retrieve the simulation results back

Job transferring and monitoring

Visual Effects of the Results

Results solved by MATLAB

Results solved by PSILAB

Correctness

In the same iteration step, at the same integration

point, the values of the small displacements on the top

of the soundboard solved by Matlab and PSILAB are:

• in the same order (10-10);

• the first three digitals of the values are almost the same

Related Works

PDE Toolbox (MathWorks) • Only supports 1D and 2D PDE-based problems;

Geodise (University of Southampton) • Uses MATLAB as a scripting language engine; • Users need to program MATLAB.

FEMLAB (COMSOL Inc.) • Does not support Grid access

GridPSi • Supports real-world 2D and 3D PDE-based problems; • Automatic generation MATLAB script without programming; • Grid accessible.

Conclusions

• Layered architecture and component-based design makes our system easily interface to Matlab.

• Automatic Matlab FEM code generation greatly reduces users’ work and programming errors.

• Simple Grid access pattern lowers barriers to enter for users who wish to run their physical simulations on the Grid.

Future Work

• The real-time visualization through Unicore/VISIT and

AVS/Express

• The design of nonlinear PDE Wizards

Thank You !