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Analytic FEM.wxm
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/* [wxMaxima batch file version 1] [ DO NOT EDIT BY HAND! ]*/
/* [ Created with wxMaxima version 18.02.0 ] */
/* [wxMaxima: title start ]
Finite Element Method
[wxMaxima: title end ] */
/* [wxMaxima: comment start ]
This is a FEM implementation in a CAS and is intented to be useful for understand the implementation of the method.
Author: Nicolas Guarin
Last rev: February, 2019
[wxMaxima: comment end ] */
/* [wxMaxima: input start ] */
kill(all);
ratprint: false$
fpprintprec: 6$
/* [wxMaxima: input end ] */
/* [wxMaxima: section start ]
Input phase
[wxMaxima: section end ] */
/* [wxMaxima: comment start ]
Mesh info
[wxMaxima: comment end ] */
/* [wxMaxima: input start ] */
coords: [
[0.000, 0.035], [0.000, 0.030], [0.000, 0.025], [0.000, 0.020], [0.000, 0.015],
[0.000, 0.010], [0.000, 0.005], [0.040, 0.035], [0.040, 0.030], [0.040, 0.025],
[0.040, 0.020], [0.040, 0.015], [0.040, 0.010], [0.040, 0.005], [0.005, 0.000],
[0.010, 0.000], [0.015, 0.000], [0.020, 0.000], [0.025, 0.000], [0.030, 0.000],
[0.035, 0.000], [0.005, 0.040], [0.010, 0.040], [0.015, 0.040], [0.020, 0.040],
[0.025, 0.040], [0.030, 0.040], [0.035, 0.040], [0.000, 0.000], [0.040, 0.000],
[0.000, 0.040], [0.040, 0.040], [0.005, 0.030], [0.005, 0.020], [0.005, 0.010],
[0.035, 0.030], [0.035, 0.020], [0.035, 0.010], [0.015, 0.010], [0.025, 0.010],
[0.015, 0.030], [0.025, 0.030], [0.010, 0.010], [0.030, 0.010], [0.010, 0.030],
[0.030, 0.030], [0.010, 0.035], [0.010, 0.025], [0.010, 0.015], [0.010, 0.005],
[0.030, 0.035], [0.030, 0.025], [0.030, 0.015], [0.030, 0.005], [0.020, 0.005],
[0.020, 0.035], [0.010, 0.020], [0.030, 0.020], [0.020, 0.010], [0.020, 0.030],
[0.015, 0.020], [0.025, 0.020], [0.020, 0.015], [0.020, 0.025], [0.020, 0.020]
]$
/* [wxMaxima: input end ] */
/* [wxMaxima: input start ] */
elements: [
[29, 16, 43, 6, 15, 50, 35, 7],
[16, 18, 19, 43, 17, 55, 39, 50],
[18, 20, 44, 59, 19, 54, 40, 55],
[20, 30, 13, 44, 21, 14, 38, 54],
[6, 43, 57, 4, 35, 49, 34, 5],
[43, 59, 65, 57, 39, 63, 61, 49],
[59, 38, 58, 65, 40, 53, 62, 63],
[44, 13, 11, 58, 38, 12, 37, 53],
[4, 57, 45, 2, 34, 48, 33, 3],
[57, 65, 60, 45, 61, 64, 41, 48],
[65, 58, 46, 60, 62, 52, 42, 64],
[58, 11, 9, 46, 37, 10, 36, 52],
[2, 45, 23, 31, 33, 47, 22, 1],
[45, 60, 25, 23, 41, 56, 24, 47],
[60, 46, 27, 25, 42, 51, 26, 56],
[46, 9, 32, 27, 36, 8, 28, 51]
]$
/* [wxMaxima: input end ] */
/* [wxMaxima: comment start ]
Element properties
[wxMaxima: comment end ] */
/* [wxMaxima: input start ] */
elprop: [2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2]$
/* [wxMaxima: input end ] */
/* [wxMaxima: comment start ]
Analysis type
1 - Plane Strain
2 - Plane Stress
[wxMaxima: comment end ] */
/* [wxMaxima: input start ] */
type:1;
/* [wxMaxima: input end ] */
/* [wxMaxima: comment start ]
Material properties
[wxMaxima: comment end ] */
/* [wxMaxima: input start ] */
En: [9.21e10, 7.31e10];
nun: [0.33, 0.325];
rhon: [8270, 2770];
t: 0.025;
/* [wxMaxima: input end ] */
/* [wxMaxima: section start ]
Common routines
[wxMaxima: section end ] */
/* [wxMaxima: comment start ]
Constitutive model tensor in Voigt notation
[wxMaxima: comment end ] */
/* [wxMaxima: input start ] */
if type=1 then(
C: E*(1-nu)/(1+nu)/(1-2*nu)*
matrix(
[1, nu/(1-nu), 0],
[nu/(1-nu), 1, 0],
[0, 0, (1-2*nu)/2/(1-nu)]
)
)else(
C: t*E/(1-nu^2)*
matrix(
[1, nu, 0],
[nu, 1, 0],
[0, 0, (1-nu)/2]
),
rhon:rhon*t
);
/* [wxMaxima: input end ] */
/* [wxMaxima: comment start ]
Spy the elements in a matrix
[wxMaxima: comment end ] */
/* [wxMaxima: input start ] */
spy(mat):= wxdraw2d(dimensions=[400,400],
colorbox=false, palette=gray,
image(signum(-abs(mat)), 0, 0, matrix_size(mat)[1], matrix_size(mat)[2])
)$
/* [wxMaxima: input end ] */
/* [wxMaxima: comment start ]
Define the Gauss points and weights
[wxMaxima: comment end ] */
/* [wxMaxima: input start ] */
gauss_pts(n):= block([pol,dpol,x1,dpoln,w1],
pol: ratsimp(legendre_p(n, x)),
dpol: ratsimp((n*legendre_p(n-1,x)-n*legendre_p(n,x)*x)/(1-x^2)),
x1:sort(map(rhs,allroots(pol))),
dpoln:subst([x=x1],dpol),
w1:makelist(2/((1-x1[k]^2)*dpoln[k]^2),k,1,n),
[x1,w1]
)$
/* [wxMaxima: input end ] */
/* [wxMaxima: comment start ]
Numerical integration
[wxMaxima: comment end ] */
/* [wxMaxima: input start ] */
gauss_int2d(f,x,y,n):=block([acu],
acu : 0,
gpts : gauss_pts(n),
for i:1 thru n do(
for j:1 thru n do(
fn : subst([x=gpts[1][i],y=gpts[1][j]],f),
acu : acu + fn*gpts[2][i]*gpts[2][j]
)
),
acu
)$
/* [wxMaxima: input end ] */
/* [wxMaxima: comment start ]
Solve linear system using Gauss' Elimination
[wxMaxima: comment end ] */
/* [wxMaxima: input start ] */
gauss_solve(A,b):=block([Aaug,Up,acu],
n:matrix_size(A)[1],
Aaug: genmatrix(lambda([i,j], 0), n, n+1),
for i:1 thru matrix_size(A)[1] do(
for j:1 thru matrix_size(A)[2]+1 do(
if j<=matrix_size(A)[2] then
Aaug[i,j]:A[i,j]
else
Aaug[i,j]:b[i,1]
)
),
Up:echelon(Aaug),
x:b*0,
for i:n step -1 thru 1 do(
acu:0,
for j:i+1 thru n do(
acu: acu + Up[i,j]*x[j,1]
),
x[i,1]: Up[i,n+1] - acu
),
float(x)
)$
/* [wxMaxima: input end ] */
/* [wxMaxima: section start ]
Elemental routines
[wxMaxima: section end ] */
/* [wxMaxima: comment start ]
Interpolation functions
[wxMaxima: comment end ] */
/* [wxMaxima: input start ] */
Haux:1/2*[(1-r^2)*(1+s),(1-s^2)*(1-r),(1-r^2)*(1-s),(1-s^2)*(1+r)]$
N:[1/4*(1+r)*(1+s)-1/2*Haux[1]-1/2*Haux[4],
1/4*(1-r)*(1+s)-1/2*Haux[1]-1/2*Haux[2],
1/4*(1-r)*(1-s)-1/2*Haux[2]-1/2*Haux[3],
1/4*(1+r)*(1-s)-1/2*Haux[3]-1/2*Haux[4],
Haux[1],Haux[2],Haux[3],Haux[4]]$
kill(Haux)$
/* [wxMaxima: input end ] */
/* [wxMaxima: comment start ]
Interpolation matrix
[wxMaxima: comment end ] */
/* [wxMaxima: input start ] */
H: genmatrix(lambda([i,j], 0), 2, 16)$
for i:1 thru 8 do(
H[1,2*i-1]:N[i],
H[2,2*i]:N[i]
)$
/* [wxMaxima: input end ] */
/* [wxMaxima: comment start ]
Local derivatives
[wxMaxima: comment end ] */
/* [wxMaxima: input start ] */
dHdr:genmatrix(lambda([i,j], 0), 2, 8)$
for i:1 thru 8 do(
dHdr[1,i]:diff(N[i],r),
dHdr[2,i]:diff(N[i],s)
)$
dHdr$
/* [wxMaxima: input end ] */
/* [wxMaxima: comment start ]
Jacobian functions
[wxMaxima: comment end ] */
/* [wxMaxima: input start ] */
jaco(dHdr,coordEl):=ratsimp(dHdr.coordEl)$
jacoinv(dHdr,coordEl):=block(
jac:jaco(dHdr,coordEl),
1/determinant(jac)*matrix([jac[2,2],-jac[1,2]],[-jac[2,1],jac[1,1]])
)$
detjac(dHdr,coordEl):=block([jac],
jac:jaco(dHdr,coordEl),
determinant(jac)
)$
/* [wxMaxima: input end ] */
/* [wxMaxima: comment start ]
Local mass matrix function
[wxMaxima: comment end ] */
/* [wxMaxima: input start ] */
local_mass(H, dHdr, coordEl, rhon):=block(
dHdx: ratsimp(jacoinv(dHdr,coordEl).dHdr),
M: rhon*transpose(H).H,
det: detjac(dHdr,coordEl),
for i:1 thru 16 do(
for j:1 thru 16 do(
integrand: det*M[i][j],
M[i][j]: gauss_int2d(integrand,r,s,5)
)
),
M
)$
/* [wxMaxima: input end ] */
/* [wxMaxima: comment start ]
Local stiffness matrix function
[wxMaxima: comment end ] */
/* [wxMaxima: input start ] */
local_stiff(dHdr,coordEl,Cn):=block([dHdx,B],
dHdx:ratsimp(jacoinv(dHdr,coordEl).dHdr),
B: genmatrix(lambda([i,j], 0), 3, 16),
for i:1 thru 8 do(
B[1,2*i-1]:dHdx[1,i],
B[2,2*i]: dHdx[2,i],
B[3,2*i-1]:dHdx[2,i],
B[3,2*i]: dHdx[1,i]
),
K:transpose(B).Cn.B,
det:detjac(dHdr,coordEl),
for i:1 thru 16 do(
for j:1 thru 16 do(
integrand:det*K[i][j],
K[i][j]:gauss_int2d(integrand,r,s,5)
)
),
K
)$
/* [wxMaxima: input end ] */
/* [wxMaxima: comment start ]
Draw an element
[wxMaxima: comment end ] */
/* [wxMaxima: input start ] */
draw_elem(nel, coords, elements):= block([el, pts, conec],
el: elements[nel],
pts: [
[coords[el[1]][1],coords[el[1]][2]], [coords[el[5]][1],coords[el[5]][2]],
[coords[el[2]][1],coords[el[2]][2]], [coords[el[6]][1],coords[el[6]][2]],
[coords[el[3]][1],coords[el[3]][2]], [coords[el[7]][1],coords[el[7]][2]],
[coords[el[4]][1],coords[el[4]][2]], [coords[el[8]][1],coords[el[8]][2]]
],
conec: [
[coords[el[1]][1],coords[el[1]][2]], [coords[el[5]][1],coords[el[5]][2]],
[coords[el[2]][1],coords[el[2]][2]], [coords[el[6]][1],coords[el[6]][2]],
[coords[el[3]][1],coords[el[3]][2]], [coords[el[7]][1],coords[el[7]][2]],
[coords[el[4]][1],coords[el[4]][2]], [coords[el[8]][1],coords[el[8]][2]],
[coords[el[1]][1],coords[el[1]][2]]
],
wxdraw2d(
color=black,
points_joined = true,
point_size = 0,
points(conec),
color=blue,
points_joined = false,
point_size = 2,
point_type = 7,
points(pts)
)
)$
/* [wxMaxima: input end ] */
/* [wxMaxima: section start ]
Assembly routines
[wxMaxima: section end ] */
/* [wxMaxima: input start ] */
KG: genmatrix(lambda([i,j], 0), 2*length(coords), 2*length(coords))$
MG: genmatrix(lambda([i,j], 0), 2*length(coords), 2*length(coords))$
/* [wxMaxima: input end ] */
/* [wxMaxima: comment start ]
Assembling...
[wxMaxima: comment end ] */
/* [wxMaxima: input start ] */
for el:1 thru length(elements) do(
coordEl: matrix(
[coords[elements[el][1]][1],coords[elements[el][1]][2]],
[coords[elements[el][2]][1],coords[elements[el][2]][2]],
[coords[elements[el][3]][1],coords[elements[el][3]][2]],
[coords[elements[el][4]][1],coords[elements[el][4]][2]],
[coords[elements[el][5]][1],coords[elements[el][5]][2]],
[coords[elements[el][6]][1],coords[elements[el][6]][2]],
[coords[elements[el][7]][1],coords[elements[el][7]][2]],
[coords[elements[el][8]][1],coords[elements[el][8]][2]]
),
mat: elprop[el],
Cn: subst([E=En[mat],nu=nun[mat]],C),
KL: local_stiff(dHdr,coordEl,Cn),
ML: local_mass(H,dHdr,coordEl,rhon[mat]),
for i:1 thru 8 do(
for j:1 thru 8 do(
ig: elements[el][i],
jg: elements[el][j],
KG[2*ig-1][2*jg-1]: KG[2*ig-1][2*jg-1] + KL[2*i-1][2*j-1],
KG[2*ig][2*jg-1]: KG[2*ig][2*jg-1] + KL[2*i][2*j-1],
KG[2*ig-1][2*jg]: KG[2*ig-1][2*jg] + KL[2*i-1][2*j],
KG[2*ig][2*jg]: KG[2*ig][2*jg] + KL[2*i][2*j],
MG[2*ig-1][2*jg-1]: MG[2*ig-1][2*jg-1] + ML[2*i-1][2*j-1],
MG[2*ig][2*jg-1]: MG[2*ig][2*jg-1] + ML[2*i][2*j-1],
MG[2*ig-1][2*jg]: MG[2*ig-1][2*jg] + ML[2*i-1][2*j],
MG[2*ig][2*jg]: MG[2*ig][2*jg] + ML[2*i][2*j]
)
)
);
/* [wxMaxima: input end ] */
/* [wxMaxima: section start ]
Solution
[wxMaxima: section end ] */
/* [wxMaxima: input start ] */
kill(KL);
/* [wxMaxima: input end ] */
/* [wxMaxima: input start ] */
kill(ML);
/* [wxMaxima: input end ] */
/* [wxMaxima: input start ] */
kill(coords);
/* [wxMaxima: input end ] */
/* [wxMaxima: input start ] */
kill(elements);
/* [wxMaxima: input end ] */
/* [wxMaxima: comment start ]
Still pending
[wxMaxima: comment end ] */
/* [wxMaxima: input start ] */
spy(KG);
/* [wxMaxima: input end ] */
/* [wxMaxima: input start ] */
spy(MG);
/* [wxMaxima: input end ] */
/* Old versions of Maxima abort on loading files that end in a comment. */
"Created with wxMaxima 18.02.0"$