mat femingles

7/27/2019 MAT FemIngles

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F. Zrate, E. Oate

CIMNE 2007


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Index:

Introduction 3Data files 4

MAT-fem 7Start 7

Elemental stiffness matrix and it assemble 8Details about the stiffness matrix 10External loads 13Fixed displacements 14Solution of the equation system 14

Reactions 15Stresses 15Writing for Postprocessing 15

About the stress calculus 16

Graphical User Interface 21Process 24Postprocess 24Example 25

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Introduction.

As any other numerical method the Finite Element Method is linked to the

programming algorithms that they define it. In fact, its advantage is in the iteration of

the same calculations on a great amount of elements. Historically the first used

programming language to work with the Finite Element Method was FORTRAN; since

then many routines, algorithms and programs associated to the method have been

programmed in this language. With the development of the computers, new languages

have appeared, each one with capacities and specific tools for diverse fields of

application, all it with the idea to simplify the codification of the algorithms and to

optimize the computer resources.

Although FORTRAN continues being a language of reference in the use of the FEM,

the new languages and programming tools allow the coding simplification in thealgorithms implementation. At the same time uses specific libraries and resources that

optimize the memory and computer resources. This is the tactical mission of MATLAB

that besides to be an investigation tool, allows us to write codes that it interprets at the

moment of the execution. From an optimal programming point of view, the interpretive

languages are very slow, however, MATLAB allows us to make use of all the

implemented matrix routines which allows optimizing the calculations until the point to

compete efficiently with other compiled languages.

MATLAB is a designed tool to work with matrices, facilitating all the matrix algebra,

from the numerical and storage point of view, giving a simple and easy way to handle

complex routines.

Having an efficient calculus program is not the only requirement to be able to work with

the Finite Element Method. It is necessary to count on a suitable interface to prepare the

problem data, to generate meshes adapted to the kind of problem to solve and to display

the results so that the interpretation of these is the main objective. MATLAB is very

efficient in treatment of matrices but very poor program in their graphical capacities.

The ideal complement is the pre/postprocessor program GiD.

GiD is a tool designed to treat any geometry as a CAD and to allocate the materials

properties and boundary conditions of the original problem. Different efforts as the

discretization and data writing in an established format become a practically transparenttask for the user.

The data processing by means of GiD is a simple task, in where the visualization of the

obtained results allows concentrating itself in the interpretation of the results.

MAT-fem has been written thinking about the joint interaction of GiD with MATLAB.

GiD allows manipulating geometries and discretizations, writing the file as MATLAB

need it. Through MATLAB the calculation program is executed, without losing the

MATLAB advantages. Finally GiD gathers the output data files for its graphical

visualization and interpretation.

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This scheme allows understanding in detail the execution a Finite Elements program,

following step by step each one of the code lines if it is desired, to the time that is

possible to heighten examples that by their dimensions would fall outside any program

with educative aims.

Este esquema permite conocer en detalle la ejecucin de un programa de ElementosFinitos, siguiendo, si se desea, paso a paso cada una de las lneas del cdigo, al tiempo

que es posible realzar ejemplos que por sus dimensiones caeran fuera de cualquier

programa con fines educativos.

In the following sections the program is described in detail. It starts with the input data

file, automatically generate by GiD, but it contains information important to understand

the operation of MAT-fem.

Finally the user interface implemented in GiD is described using an application

example.

Data Files.

Before describing the program it is obvious to say that somehow it is necessary to feed

it with the information corresponding to the nodal coordinates, the element

discretization and the boundary conditions. For that reason, the input data file is firstly

described in order to familiarize with the programming style and the variables used in

MATLAB. As we commented, MATLAB is an interpreter and we will use this property

to define the input data. That is to say, that the input data file is in fact one subroutine of

the program in where the values corresponding to the problem are assigned directly to

the variables.

It avoids having to define a special reading syntax for the program and at the time it also

avoids us to implement an I/O interface, concentrating in programming the method.

Therefore, the input data file will use the MATLAB syntax. The variables used by the

program are defined directly in it. It is obvious that the name of this file will have to

take the MATLAB extension .m

Inside the data file we can distinguish three groups of variables: the associates to the

material; those that define the topology of the problem and those that define theboundary conditions. With the intention of simplifying the code, an isotropic linear

elastic material for the whole domain is used, due to that the material section appears

once in the data file.

Figure 1 shows the variables associated to the material: the variable pstrs indicates if it

is a problem of plane stress (pst r s = 1) or plane strain (pst r s = 0). young contains

the elasticity modulus andpoi ss the Poissons coefficient. t hi ck anddenss define the

thickness and the density of the domain respective. It is obvious that in a plan strain

problem the thickness value must be unitary.

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case of using quadrilateral elements. Similar to the previous variable, the number of

element corresponds with the number of row that keeps in the defined matrix.

The last group of variables defines the boundary conditions of the problem, as it is

shown in figure 3.

Figure 3: Input data file: Boundary conditions definition

%% Fi xed Nodes%fixnodes = [

1, 1, 0.0 ;1, 2, 0.0 ;

13, 1, 0.0 ;13, 2, 0.0 ];

%% Poi nt l oads%

pointload = [

6, 2, -1.0 ;

18, 2, -1.0 ];

%% Si de l oads%sideload = [

11,12, 2.0, 3.0;

14,15, 2.0, 3.0 ];

The variable f i xnodes corresponds to the degrees of freedom (DOF) restricted in

agreement with the problem to solve. f i xnodes is a matrix where the number of rowscorresponds to the number of prescribed DOF and the number of columns describes in

the following order the restricted node, the fixed DOF code (1 in x direction and2 in y

direction) and the value for this DOF. In this way if a node is prescribed in both

directions two lines are necessary to define this condition.

The poi nt l oad variable is used to define the punctual loads. Like the previous

variables, this is a matrix where the number of rows is the number of defined loads in

the problem and the number of columns corresponds to the number of the loaded node,

the direction in which acts and the value of the load. The loads are referred in the global

system of coordinates. In the case of not existing punctual loads poi nt l oad is defined

as an empty matrix by means of the commandpoi nt l oad = [ ] ;

Finally the variable si del oad contains the information of the loads uniformly

distributed throughout the sides of the elements. The definition of the loads is in the

global reference system. si del oad is a matrix where the number of rows is the number

of element sides with this type of load and the two first columns define the nodes of this

side while columns 3 and 4 correspond to the value of the load distributed by unit of

length in direction x andy respectively. If there is no uniform loads sideload must be

defined as an empty matrix by means of the commandsi del oad = [ ] ;

The name of the data file is up to the user; nevertheless, the extension must be .m inorder that MATLAB can recognize it.

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MAT-fem

MAT-fem is a TOP-DOWN execution program. The program flow chart is shown in

Figure 4. The first part, the input data section, is made in the same file were the data isdefined as it was described in the previous section.

The next task is to evaluate and assemble the elemental stiffness matrix and the

elemental mass load vector

Start

Read input data *.m

Figure 4: MAT-fem flow chart

In the same figure, before eliminating the equations related to the known DOF and

solving the equation system, all the load conditions are applied to the load vector.

Once the unknown DOF are found, the program evaluates the systems reactions and

stresses, finalizing with the data writing to visualize them in GiD.

Compute K(e)

and f(e)

Define f(e) for point and

distributed loads along a side

Solve Ka=f

Compute node reactionsR = Ka - f

Compute smoothed

stresses at nodes

Write results for GiD

Assembling of K(e) and f(e)

Function TrStiffor trianglesFunction QdStinffor quadrilaterals

Function Stress:

Call function TrStrs for trianglesCall function QsStrs for

uadrilaterals

Compute D Function constt

Loop over elements

Function ToGiD

End

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Start

AT-fem begins cleaning all the variables with the cl ear command. Next it asks to the

Figure 5: Program initia zation and data reading

he data reading, as it was said before, is a direct variable allocation in the program.

he total number of equations per element will be the number of nodes by element

is interesting to reflect that these variables are not strictly necessary since they are

hroughout the program the t i mi ng routine is used to calculate the run time between

M

user the name of the input data file that is going to use (the .m extension in not included

in the filename). Figure 5 shows the first lines of the code corresponding to thevariables boot as well as the clock set up, storing in the variable t t i mthe total time of

execution.

%% MAT- f em

r y and var i abl es.

me = input('Enter the file name :','s');

me count er

,1); % Number of nodesOF

l ement

%Cl ear memo%

clear

file_na

tic; % St art c l ockttim = 0; % I ni t i al i ze t i

eval (file_name); % Read i nput f i l e

Fi nds basi cs di ment i ons% npnod = size(coordinates nndof = 2*npnod; % Number of t otal D nelem = size(elements,1); % Number of el ements

e nnode = size(elements,2); % Number of nodes per neleq = nnode*2; % Number of DOF per el ement

ttim = timing('Time needed to read the input file',ttim);

li

TFrom these matrices it is possible to extract the basic dimensions of the problem such as

the number of nodal points npnod which corresponds to the number of lines in the

coor di nat es matrix. The number of total degrees of freedom of the program nndof

will be twice the previous amount (2*npnod). nel emcorresponds to the number of read

elements and is the number of lines contained in the matrix el ement s whereas the

number of nodes that contain each elements is the number of columns of this matrix; in

this way the triangular elements are identified if the number of columns is three whereas

for the quadrilateral elements the number of columns will be of four.

T

nnode multiplied by the number of DOF for each node: two in the case of bidimensional

problems.

It

defined in the data structure, nevertheless, their definition simplify the code

interpretation.

T

two points in the code. In this way the user can observe the programs subroutine that

requires higher computational effort. Inside t i mi ng the t i c and t oc MATLAB

commands are used.

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Elemental stiffness matrix and it assemble.

he code shown in Figure 6 respectively defines the global stiffness matrix and the

Figure 6: Global matrix initialization

ince the programs main purpose is to demonstrate the implementation of the FEM

he subroutine const t starts with the Youngs modulus, the Poissons coefficient and a

x definition.

this program it has been decided to recalculate values instead of storing them. The

igure 8 shows the elemental loop in which the program calculates and assembles the

T

equivalent nodales forces vector as a spar se matrix and vector. MAT-fem uses sparse

matrices to optimize the memory using the MATLABs tools. On this way and without

additional effort MAT-fem makes use of very powerful algorithms without losing thewriting simplicity.

% Di mensi on t he gl obal mat r i ces.); % The gl obal st i f f ness mat r i xStifMat = sparse ( nndof , nndof

force = sparse ( nndof , 1 ); % The gl obal f orce vect or

Mater i al pr oper t i es ( Const ant over t he domai n) .% dmat = constt(young,poiss,pstrs);

S

some simplifications are made like using a unique material in the whole domain, in this

way the constitutive matrix will not vary element to element and it is evaluated before

initiating the element stiffness matrix.

T

flag that allows us to distinguish between a plane stress or plane strain problem. The

constitutive matrix is stored in dmat as it is observed in Figure 6. In Figure 7 the

subroutine const t shows the explicit form for the dmat for an isotropic linear elastic

material.

Figure 7: Constitut e matri

function D = constt (young,poiss,pstrs)

g/(1-poiss^2);

ng*(1-poiss)/(1+poiss)/(1-2*poiss);

[aux1,aux2,0;aux2,aux1,0;0,0,aux3];

Pl ane Sr ess%

if (pstrs==1)aux1 = younaux2 = poiss*aux1;

oiss);aux3 = young/2/(1+p

% Pl ane St r ai nelse

ux1 = youaaux2 = aux1*poiss/(1-poiss);aux3 = young/2/(1+poiss);thick= 1.0;

end

=D

iv

In

computers speed makes this without reduction of the program efficiency. In addition, it

allows having more memory to attack greater problems.

F

stiffness matrix and the equivalent nodal load vector for each element. The cycle begins

recovering the geometric properties of each element. In the vector l nods the elements

nodal connectivity are stored and in the coord matrix the coordinates from these nodes

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are keep.

Figure 8: Elemental stiffness m nd assembling.

the next step the elemental stiffness matrix is calculate. Depending on the element the

efore making the assembly the eqnum vector is defined. It contains the global

he assembly is made by means of two cycles from 1 t onel eq (number of equations

etails about the stiffness matrix

this program two finite elements are used the triangular whose stiffness matrix is

% El ement cycl e.elem

i es% conecti vi t i es

gl e

he equati on number l i st f or t he i - t h el ementr t he l i s t

on

eqnum(j)) + ...

End el ement ci cl e

for ielem = 1 : n

Recover el ement proper t%

lnods = elements(ielem,:); coord(1:nnode,:) = coordinates(lnods(1:nnode),:); % coor di nat es

Eval uates t he el ement al st i f f nes matr i x and mass f orce vect or.%i f (nnode == 3)

or] = TrStif(coord,dmat ,thick,denss); %Tr i an[ElemMat,ElemF else

mMat,ElemFor] = QdStif(coord,dmat ,thick,denss); % Quad.[Ele end

Fi nds t% eqnum = []; % Cl ea for i =1 : nnode % Node ci cl e

quat i eqnum = [eqnum,lnods(i)*2-1,lnods(i)*2]; % Bui l d t he e end % number l i st

Assambl e the f or ce vect or and t he st i f f nes matr i x% for i = 1 : neleq

= force(eqnum(i)) + ElemFor(i);force(eqnum(i))for j = 1 : neleq

),eqnum(j)) = StifMat(eqnum(i),StifMat(eqnum(iElemMat(i,j);

endend

end %

atrix evaluation a

In

subroutine Tr St i f is called for triangular elements and QdSt i f for quadrilateral

elements. The same subroutine evaluates the element stiffness matrix and the element

load vector. The use of the same area integration allows this simplification. The

calculation of the elemental stiffness matrices requires a detailed study made in the

following section.

B

equations number for each one of the equations in the elemental stiffness matrix. The

number conversion is simple because for each node two equations corresponds (one byeach degree of freedom).

T

in each element) assembling in the first cycle the equivalent nodal load vector and in the

second the elemental stiffness matrix term by term. This scheme avoids storing the

element matrices temporarily.

D

In

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calculates in an explicit form and the quadrilateral element calculates by means of

numerical integration. Both routines require exactly the same input data and also return

the same variables: El emMat for the stiffness matrix and El emFor for the nodal

equivalent mass load vector.

TheTr St i f subroutine is defined for triangular elements as it is shown in Figure 9. In it

he elemental stiffness matrix is calculates by means of the classic expression T

he equivalent nodal mass load vector is evaluated distributing equal l y the weight of

Figure 9: Elemental stiffness matrix for the triangular element.

or the quadrilateral element the evaluation of the stiffness matrix is made using a

t this level also the Gaussian-Legendre coordinates values (pospg) and the weights

the Cartesian derivatives of the triangular linear function forms are calculate directly.

Those derivates are constant in the entire element. In this way the bmat matrix is built

simply placing each one of the Cartesian derivatives (b( i ) / ar ea2 andc( i ) / area2) in

the corresponding position of the deformation matrix, without forgetting the common

factor1/ area2.

T B DB

dA. In this line one of the fundamental advantages of MATLAB is observed where themultiplication of matrices is direct with no need to write troublesome cycles.

T

the element in each node (f orce). This load represents a negative force in y direction.

function [M,F] = TrStif ( nodes,dmat,thick,denss)

k

% ci = xk - xj

);

(2), 0 ,b(3), 0 ; % Expl i c i t f orm f or mat ri x B

mat*bmat)*area*thick; % El ement al st i f f nes mat r i x

ce]; % El ement al f orce vect or

b(1) = nodes(2,2) - nodes(3,2); % bi = yj - yb(2) = nodes(3,2) - nodes(1,2);

b(3) = nodes(1,2) - nodes(2,2);

c(1) = nodes(3,1) - nodes(2,1);c(2) = nodes(1,1) - nodes(3,1);c(3) = nodes(2,1) - nodes(1,1);

area2 = abs(b(1)*c(2) - b(2)*c(1)area = area2 / 2;

bmat = [b(1), 0 ,b0 ,c(1), 0 ,c(2), 0 ,c(3);

c(1),b(1),c(2),b(2),c(3),b(3)];bmat = bmat / area2;

M = (transpose(bmat)*dforce = area*denss*thick/3;F = [0,-force,0,-force,0,-for

F

numerical integration. Initially the elements function forms and its natural derivatives

are defined (f f o r m and der i v). This definition is made by means of an intrinsic

function. This facility allowed in MATLAB avoids the use of additional subroutines.

A

(pespg) corresponding to the 2x2 integration rule are defined. Also the boot of the M

matrix for the elemental stiffness matrix and the equivalent nodal load vectorf y is made

on this section. In figure 10 this routine is observed with detail.

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Once the variables are initialized two loops defines the integration along the global

Figure 10: Elemental stiffness matrix for the quadrilateral element.

he deformation matrix is obtained placing properly the Cartesian derivates of the

he equivalent nodal mass load vector also requires integrate over the elements area

o finalize the routine in the F vector all the nodal forces are properly placed having in

axes. The l c f f m vector contains the values of the function form evaluated at the

integration point i , j and the l cder matrix stores the values for the natural derivative

of these functions at the same integration point. The Jacobian matrix (xcj acm) is

evaluated by the simple multiplication of the l cder matrix by the nodales coordinates

of the element. The value of the Cartesian derivates (ct der) in the integration point isobtained multiplying the inverse of the Jacobian matrix by the natural derivates of the

function form. The area differential is not another thing more than the determinant of

the Jacobian matrix by the weight functions of the integration point without forgetting

the element thickness.

function [M,F] = QdStif ( nodes,dmat,thick,denss)

+t+s*t)/4,(1-s+t-s*t)/4];

0 ];

2orm(pospg(i),pospg(j)) ; % FF at gauss poi nt

s

1 : 4 ctder(1,inode), 0 ;

;

+ (transpose(bmat)*dmat*bmat)*darea;

0, -fy(1), 0, -fy(2), 0, -fy(3), 0, -fy(4)];

fform = @(s,t)[(1-s-t+s*t)/4,(1+s-t-s*t)/4,(1+sderiv = @(s,t)[(-1+t)/4,( 1-t)/4,( 1+t)/4,(-1-t)/4 ;

(-1+s)/4,(-1-s)/4,( 1+s)/4,( 1-s)/4 ];

pospg = [ -0.577350269189626E+00 , 0.577350269189626E+0pespg = [ 1.0E+00 , 1.0E+00];

M = zeros(8,8);fy = zeros(1,4);

or i=1 : 2ffor j=1 :

lcffm = ff lcder = deriv(pospg(i),pospg(j)) ; % FF Local der i vat i ve xjacm = lcder*nodes ; % J acobi an mat r i x

i vat es ctder = xjacm\lcder ; % FF Cart esi an der darea = det(xjacm)*pespg(i)*pespg(j)*thick;

mat = [];b

for inode =bmat = [ bmat , [0 ,ctder(2,inode);

ctder(2,inode),ctder(1,inode) ] ]end

= MM

y = fy + lcffm*denss*darea;f

endend

F = [

T

function form in a matrix array, to obtain by means of the classic BTDB expression the

elemental stiffness matrix. Do not forget that to make a numerical integration it is

necessary to evaluate the sum of the variables calculated in all of the Gaussian points.

This sum is made in the same matrix M

T

multiply the function form by the element density.

T

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mind that only the negative y force component exists.

These two routines try to exemplify the two mechanisms for the stiffness matrix

xternal loads.

esides the mass load it is necessary to consider uniformly loads distributed on the

ince both implemented elements have linear functions form, the calculation of the

bserve that the loads are defined in the global coordinate system and it is not

Figure 11: Equivalent nodal force vector for a uniform distributed load.

or the punctual loads, the calculation is as simple as adding the value of the load to the

Figure 12: Equivalent nodal force vector for punctual load.

calculation using the explicit form or a numerical integration.

E

B

sides of the elements and punctual loads in the nodes.

S

nodal contribution for the uniformly distributed loads will be exactly the same for both

cases. The evaluation is made after the assembly of the stiffness matrix, in the main

routine of the MAT-fem program. This code appears in Figure 11 where there is a loop

over the number of loads defined by si del oad. The nodal contribution is evaluated

equality on each node (due the linear function form).

O

necessary to make any rotation or transformation.

% Add si de f orces to t he f orce vect or

dinates(sideload(i,2),:);

node

de

for i = 1 : size(sideload,1)),:)-coorx=coordinates(sideload(i,1

l = sqrt(x*transpose(x)); % Fi nds t he l enght of t he si de ieqn = sideload(i,1)*2; % Fi nds eq. number f or t he f i r st

force(ieqn-1) = force(ieqn-1) + l*sideload(i,3)/2; % add x f or ceforce(ieqn ) = force(ieqn ) + l*sideload(i,4)/2; % add y f or ce

ieqn = sideload(i,2)*2; % Fi nds eq. number f or t he second noforce(ieqn-1) = force(ieqn-1) + l*sideload(i,3)/2; % add x f or ceforce(ieqn ) = force(ieqn ) + l*sideload(i,4)/2; % add y f or ce

end

F

equivalent nodal force vector. A loop over the number of point loads is necessary,

finding for each the equation number associate for adding the value of the load to the

force vector.

% Add poi nt l oads condi t i ons t o t he f orce vect or

% Fi nds eq. numberfor i = 1 : size(pointload,1)

+ pointload(i,2);ieqn = (pointload(i,1)-1)*2force(ieqn) = force(ieqn) + pointload(i,3); % add t he f orce

end

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Fixed displacements

The u vector is defined to hold the DOF of the problem. In Figure 13 the loop over the

prescribed DOF can be observed and how the values defined by the f i xnodes matrix

are assigned to the u vector. Also the f i x vector is defined to keep the equationnumbers of the restricted DOF.

Finally the f orce vector is updated with the product of the St i f Mat matrix ant the u

vector which at this moment contains only the values of those degrees of freedom which

have been restricted.

% Appl i es t he Di r i chl et cou = sparse ( nndof, 1 );

ndi t i ons and adj ust t he r i ght hand si de.

for i = 1 : size(fixnodes,1)ieqn = (fixnodes(i,1)-1)*2 + fixnodes(i,2); %Fi nds t he equat i on numberu(ieqn) = fixnodes(i,3); %and store t he sol ut i on i n ufix(i) = ieqn; % and mark t he eq as a f i x val ue

endforce = force - StifMat * u; % adj ust t he rhs wi t h t he known val ues

Figure13: Update of the force vector due to the known DOF

Solution of the equation system

The strategy used in MAT-fem consists of solving the global equation system without

considering those DOF whose values are known. The Fr eeNodes vector contains the list

of the equations to solve.

The Fr eeNodes vector is used as a DOF index and allows us to write in a simple way

the solution to the equations system. MATLAB takes care to choose the most suitable

algorithm to solve the problem, been totally transparent for the user the solution of the

system. The routines implemented in the MATLAB kernel nowadays compete in speed

and memory optimization with the best existing algorithms, reason why all the power of

calculation available is described in a simple line.

% Comput e the sol ut i on by sol vi ng St i f Mat * u = f orce f or t he

% r emai ni ng unknown val ues of u.FreeNodes = setdiff ( 1:nndof, fix ); % Fi nds t he f r ee node l i st and

% sol ve f or i t .u(FreeNodes) = StifMat(FreeNodes,FreeNodes) \ force(FreeNodes);

Figure 14: Solution of the equation system.

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Reactions

The solution to the equations system is in the u vector, therefore the reaction calculus is

made by means of the expression R = St i f Mat *u - f or ce. It is obvious that the value

of the reaction in those prescribed nodes is not null. In order to avoid unnecessarycalculations we use the vectorf i x which contains the list of the equations associated to

the fixed DOF as it is shown in Figure 15.

% Comput e the r eact i ons on t he f i xed nodes as a R = St i f Mat * u - Freaction = sparse(nndof,1);reaction(fix) = StifMat(fix,1:nndof) * u(1:nndof) - force(fix);

Figure 15: Reactions.

Stresses.

Once the nodal displacements have been found it is possible to evaluate the stresses in

the elements by means of the DBu expression. Since the deformation matrix B is

calculated at the integration points the stresses are referred to these points. In order to

transfer the values of the stresses at the integration points towards the element nodes it

is necessary to review in detail and in a later section the calculation of these values. The

Figure 16 presents the subroutine call for the nodal stresses evaluation which are store

in the St r nod matrix.

% Comput e t he st r essesStrnod = Stress(dmat,poiss,thick,pstrs,u);

Figure 16: Stress evaluation call.

Writing for Postprocessing

Once calculated the nodales displacements, the reactions and the stresses it is come to

overturn these values to the postprocess files from where GiD will be able to

present/display them in a graphical way. This is made in the subroutineToGi D shown in

Figure 17

% Gr aphi c r epr esent at i on.ToGiD (file_name,u,reaction,Strnod);

Figure 17: Postprocess call.

During the program execution will appear in the MATLAB console the total time used

by the program as well as the time consumed in each one of the described subroutines.

It is remarkable to observe that the biggest time consumption is made in the calculation

and assembly of the global stiffness matrix, whereas the solution of the equation system

really represents a small percentage of the consumed time.

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Once the program execution is finish the variables are still alive inside MATLAB

reason why it is possible to experiment with the battery of internal functions of

MATLAB.

About the stress calculus

The stress calculation in the elements requires the use of a specific subroutine, not only

for the calculus itself but also to project the stresses from the integration points towards

the nodes.

The stress subroutine controls the program flow towards the three or four nodes element

routines. In the case of the triangular element the stresses are constant in the element

and the nodal extrapolation is trivial. Not thus it is the case of the quadrilateral element

where the stresses present a bilinear variation and the stress extrapolation is made using

the same elements functions form.

Figure 18 presents the initial part of the stress subroutine were the input data are the

material constitutive matrix dmat , Poissons coefficient poi ss, the thickness thi ck, the

flag for the problem type pstrs and the nodal displacements u. Additionally the nodal

coordinates and the element connectivities will be used (defined as a global variables).

In order to simplify the reading of the routine some variables are extracted like nel emto

indicate the number of elements, nnode to indicate the number of nodes per element and

npnod for the total number of nodes in the domain.

It is important to have into account the number of stresses for plain stress problems (x,

y andxy) and plane strain (x, y, z andxy) were z is a function ofx andy

The nododst r matrix is initialized to zeros to store the value of the nodal stresses; in

the last column the number of elements that concur in the node are counted. This is

necessary to make a nodal stress mean.

function S = Stress (dmat,poiss,thick,pstrs,u)

%% St r ess Eval uates t he st r esses at t he gauss poi nt and smoot h the val ues% t o t he nodes.%global coordinates;

global elements;

nelem = size(elements,1); % Number of el ement snnode = size(elements,2); % Number of nodes per el ementnpnod = size(coordinates,1); % Number of nodes

if (pstrs==1)nstrs= 3; % Number of Str s. Sx Sy Txy

elsenstrs= 4; % Number of Str s. Sx Sy Sz Txy

endnodstr = zeros n nod nstrs+1

Figure 18: Variables boot for stress calculus...

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Like in the stiffness matrix, the stress evaluation requires a loop over the elements,

recovering the elements connectivities (l nods), coordinates for these nodes (coord) as

well as the displacements in di spl as it is shown in Figure 19

for ielem = 1 : nelem

% Recover element propertieslnods = elements(ielem,:);coord(1:nnode,:) = coordinates(lnods(1:nnode),:);eqnum = [];for i =1 : nnodeeqnum = [eqnum,lnods(i)*2-1,lnods(i)*2];

enddispl = u(eqnum);

Figure 19: Recovering of the element coordinates and nodal displacement.

Figure 20 shows the call for the stress calculus in triangular and quadrilateral elements.

In the triangular elements case the routine returns the El emSt r vector that contains the

stresses in the element; these values are accumulated in the nodst r were the last column

is the number of elements that share the node to evaluate later the nodal average.

In the quadrilateral element case the El emSt r variable represents a matrix that contains

the stresses of each one of the elements nodal points. Like in the previous case, the last

column holds the number of elements that share the node. Once all the element stresses

have been calculated and accumulated in nodst r a nodal mean is made to obtain a

smoothed stress field at the nodes.

% St r esses i nsi dif (nnode == 3)e the el ement s.

% Tr i angul ar el ement s const ant s t r essElemStr = TrStrs(coord,dmat,displ,poiss,thick,pstrs);

for j=1 : nstrsnodstr(lnods,j) = nodstr(lnods,j) + ElemStr(j);

endnodstr(lnods,nstrs+1) = nodstr(lnods,nstrs+1) + 1;

else

% Quadr i l ater al el ement s st r ess at nodesElemStr = QdStrs(coord,dmat,displ,poiss,thick,pstrs);

for j=1 : 4for i = 1 : nstrsnodstr(lnods(j),i) = nodstr(lnods(j),i) + ElemStr(i,j);

endendnodstr(lnods,nstrs+1) = nodstr(lnods,nstrs+1) + 1;

endend

% Fi nd t he mean st r ess val ue

S = [];for i = 1 : npnodS = [S ; nodstr(i,1:nstrs)/nodstr(i,nstrs+1)];end

Figure 20: Stress calculus and nodal smooth.

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The stress calculus for each element is in the following figures. For the triangular

element, the B matrix is recalculated as it was done for the stiffness matrix. The stresses

are directly the DBu product

function S = TrStrs (nodes,dmat,displ,poiss,thick,pstrs)b(1) = nodes(2,2) - nodes(3,2);b(2) = nodes(3,2) - nodes(1,2);b(3) = nodes(1,2) - nodes(2,2);

c(1) = nodes(3,1) - nodes(2,1);c(2) = nodes(1,1) - nodes(3,1);c(3) = nodes(2,1) - nodes(1,1);

area2 = abs(b(1)*c(2) - b(2)*c(1));area = area2 / 2;

bmat = [b(1), 0 ,b(2), 0 ,b(3), 0 ;

0 ,c(1), 0 ,c(2), 0 ,c(3);c(1),b(1),c(2),b(2),c(3),b(3)];

se = dmat*bmat*dis l /area2

Figure 21: Stress calculus for the triangular element.

Depending on the problem selected, the stresses that can be calculated are three for

plane stress and four for plane strain (z= - poi ss*(x + y) ) as it is shown in Figure

22.

% Pl ane Sr essif (pstrs==1)

S = se ;

% Pl ane St r ai nelse

S = [se(1),se(2),-poiss*(se(1)+se(2)),se(3)];

end

Figura 22: Plane stress or plane strain stress calculus.

Due the similarly calculus with the stiffness matrix, most of the variables used have

been described in the four nodes stiffness matrix section. Nevertheless, the most

relevant aspect is the stress extrapolation from the Gaussian points towards the nodes.

The stress variation inside the element corresponds to a linear field, therefore, we use

element functions forms to extrapolate the values to the nodes. Considered that points i

and j are in a dominion that goes from - 1/ p to +1/ p were p is the natural coordinate of

the Gaussian points, In the normalized space makes them to be a distance of - 1 and+1

as it is shown in Figure 23 a).

Another consideration consists of establishing a suitable correlation between the nodes

and the Gaussian points numbers, because the order that keep the integration points

does not agree with the element nodal numeration as it is shown in Figure 23 b).

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Like in the triangular element case, the stresses are calculate as a plain stress or plane

strain problem as it shows figure to it 24

% Pl ane Sr essif (pstrs==1)

S = se ;% Pl ane St r ai nelse

S = [se(1,:) ; se(2,:) ; -poiss*(se(1,:)+se(2,:)) ; se(3,:)];

end

Figure 24: Stress calculus for plane stress or strain problem

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The shown button is used for punctual loads allocation. When selecting, an

emergent window (Figure 26) allows giving the load value in the global

coordinate system. Once it is defined is necessary to select the nodes were to applied it.

Figure 26: Punctual load condition...

The button associated to the uniformly distributed loads permits to assign this

condition on the geometrys lines. The emergent window (Figure 27) allows

introducing the value of the load by length unit referred the global axes system.

Figure 27 Uniform load assignations.

The material properties definition is made trough the following button which

leads to a new emergent window (Figure 28) to define the material variables like

the Youngs modulus, the Poissons coefficient, density and thickness. It is necessary to

assign this material in all the areas that defines the domain. As was mention before, in

MAT-fem, for simplify reasons only one material is allowed.

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Figure 28 Material properties definition.

The general properties button allows acceding to the window shown in Figure 29

where the title of the problem is identified, the problem type (plane stress or

plane strain) the gravity loads and the units for the results.

Figure 29 General properties of the problem.

Once defined the boundary conditions and the material properties it is necessary

to make the domain discretization. The mesh button shown is used to create the

mesh with all the GiDs toolbox facilities.

The data file writing is made when pressing the last button shown in the menu.

All the geometrical properties of the problem, as well as the material are written

to the data file in the specific reading format for MAT-fem. Do not forget the .m

extension for the file name.

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Figure 30 Input data file definition.

Process.

The problem calculation is made with MATLAB. The execution does not have greater

complications than knowing in which directory is the written file. A good practice is to

set this directory the working directory were also the postprocess file will be written.

Postprocess.

Once concluded the problem execution in MATLAB, is necessary to return to GiD for

the file postprocess to analyze the obtained results. It is necessary to open any of the

generated files that contain the extension *.flavia.msh or *.flavia.res.

Figure 31 Postprocess file reading.

The obtained results visualization can be done in a diversity forms, due the GiDs

graphical possibilities which permits to show the results like a colors gradient, iso-lines,

cuts and graphs; allowing the simple interpretation of the obtained results.

In Figure 32 there is an iso-areas example showing the vertical displacement in a test

beam clamped in its left end and point loaded on its right side.

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Figure 32 Beam postprocess.

Example.

The example below is presented to show the programs simplicity of use. The problem

show corresponds to the IC6-NAFEMS benchmark were the stresses and displacement

of a circular wall subjected to a punctual load are calculated.

In this section the input data file of the proposed example is described in detail for

quadrilateral elements. Also the convergence of the results is analyzed changing the

problem discretization with triangular and quadrilateral elements.

The example is the NAFEMSS IC6 benchmark proposed in Linear Static Benchmarks

vol 1. A circular wall loaded with a punctual force. In Figure 33 the complete

geometry of the problem is presented were only one quarter is analyzed due symmetry.

The material properties are described in the same figure. The hypotheses used are the

plane strain ones.

P = 10 MN

E = 210.103 MPa

v = 0.3

t = 1.0 m

Figure 33 circular wall with punctual load. NAFEMS IC6.

The wall is symmetry fixed in both sides (the normal displacement is zero). The

punctual load is applied in the outer middle side at 45. The goal is to determine thestress x at the point A. The exact value is x = 53.2MPa

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The problem definition for MAT-fem is done using the menu described in the GUI

section. The displacement, loads, materials and problem definitions are shown in

Figures 34 and 35.

Figure 34 Boundary conditions and loads over the structure

Figure 35 Material and problem definition.

Finally the mesh discretization and the input data files are created with the last two

buttons of the MAT-fem menu.

All the meshes used are structured. For the simplest quadrilateral element case it

consists of 8 elements and 15 nodes. Figure 36 shows the nodal and element numeration

of the used mesh. In the same figure the code for the input data file is presented using

the mesh described.

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%=======================================================================% MAT- f em 1. 0 - MAT- f em i s a l ear ni ng t ool f or under st andi ng% t he Fi ni t e El ement Method wi t h MATLAB and Gi D%=======================================================================% PROBLEM TI TLE = NAFEMS I C6% Mat er i al Pr oper t i es%

young = 210103000.00000 ;poiss = 0.30000 ;denss = 0.00 ;

pstrs = 0 ;thick = 1 ;

%% Coor di nates%global coordinatescoordinates = [

11.00000 , 0.00000 ;10.50000 , 0.00000 ;10.00000 , 0.00000 ;9.23880 , 3.82683 ;9.70074 , 4.01818 ;

10.16267 , 4.20952 ;7.07107 , 7.07107 ;7.42462 , 7.42462 ;7.77817 , 7.77817 ;3.82683 , 9.23880 ;4.01818 , 9.70074 ;4.20952 , 10.16267 ;0.00000 , 10.00000 ;0.00000 , 10.50000 ;0.00000 , 11.00000 ] ;

%% El ement s%global elements

elements = [2 , 5 , 4 , 3 ;1 , 6 , 5 , 2 ;5 , 8 , 7 , 4 ;6 , 9 , 8 , 5 ;8 , 11 , 10 , 7 ;9 , 12 , 11 , 8 ;11 , 14 , 13 , 10 ;12 , 15 , 14 , 11 ] ;

%% Fi xed Nodes%fixnodes = [

1 , 2 , 0.00000 ;2 , 2 , 0.00000 ;

3 , 2 , 0.00000 ;13 , 1 , 0.00000 ;14 , 1 , 0.00000 ;15 , 1 , 0.00000 ] ;

%% Poi nt l oads%

pointload = [9 , 1 , -7071067.81200 ;9 , 2 , -7071067.81200 ] ;

%% Si de l oads%sideload = [ ];

Figure 36 Input data file for the NAFEMS IC6 benchmark problem.

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As it is observed, the input data file contains all the sections described before, the

nodales coordinates (in sequence consecutive), the material properties, coordinates (in

sequence consecutive), the element connectivities, as well as the boundary conditions,

punctual loads and the uniform loads that are defined by an empty matrix.

The program execution is made trough MATLAB with the MAT-fem command. It ispossible to observe that the greater time consumption is in the stiffness matrix assembly

due to the storage system where the internal indices of the sparse matrix must be

updated. The total running time of this problem is about 0,15 seconds.

>> MATfemEnter the file name :cilQ2Time needed to read the input file 0.003399Time needed to set initial values 0.000912Time to assamble the global system 0.032041Time for apply side and point load 0.000196Time to solve the stifness matrix 0.026293

Time to solve the nodal stresses 0.024767Time used to write the solution 0.066823

Total running time 0.154431>>

Figure 37 MATLAB console for running the NAFEMS IC6 problem.

Some of the facilities given by MATLAB are shown in figure 38 where with the

spy(StifMat) command the profile of the stiffness matrix can be visualized. With the

aid of other command it is possible to know with detail the properties of this matrix, like

the eigenvectors, the rank, its determinant, etc.

Figure 38 Stiffness matrix profile.

In Figure 39 the deformed mesh is presented as well as the x distribution inside thearc.

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Figure 39 Deformed mesh andx stress distribution

In the analyzed case the stress x in the A point reaches the value of 14,186 MPa farfrom the given solution, nevertheless, when the number of DOF are increased the target

solution is reached as it is shown in the table of the Figure 40 where the convergence of

the total displacement and the stress at the point is presented for triangles and

quadrilaterals structured meshes. The corresponding graph is in Figure 41.

TRIANGULOS CUATRILATEROS

NODOS DOF DESP x TR NODOS DOF DESP SQR x SQR

15 30 0,58607 7,8835E+06 15 30 0,80306 1,4186E+07

28 56 0,80865 9,8442E+06 28 56 1,30890 2,1405E+07

45 90 1,06980 1,2561E+07 45 90 1,79900 2,7571E+0766 132 1,34800 1,5697E+07 66 132 2,22210 3,2746E+07

120 240 1,88800 2,2193E+07 120 240 2,84280 4,0014E+07

231 462 2,54040 3,0504E+07 231 462 3,36800 4,5889E+07

1326 2652 3,73390 4,6827E+07 861 1722 3,91130 5,1518E+07

1891 3782 3,84840 4,8496E+07 1891 3782 4,03990 5,2669E+07

3321 6642 3,97260 5,0340E+07 3321 6642 4,09080 5,3056E+07

Figure 40 Convergence

1,00E+05

1,01E+07

2,01E+07

3,01E+07

4,01E+07

5,01E+07

6,01E+07

10 100 1000 10000

T

Q

Solucin : 53.2 MPa

Figure 41 Stress convergence

mat femingles

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