MaplePrimes Questions

Hello;

Hope you are fine. Can i apply numerical scheme on maple for the following problem. This in integro-differential equation i think. Waiting for kind response.

Thanks

 


could you please help me ,the maple code for this given series.

restart

U[0](x) := x;

x

(1)

"U[k+1](x):=solve((k+1)*U[k+1](x)+(x*(-1)^((k-1)/(2)))/(k!)-x^(2)*((e)^(x))/(10){6/(k!)-sum((2^(k[1]))/(k[1!])(5*delta[k[]-k[1]](x)+(2^(k-k[1])*(-1)^((k-k[1])/(2)))/((k-k[1])!)+(2^(k-k[1]+1)*(-1)^((k-k[1]-1)/(2)))/((k-k[1])!)),k[1]=0..k)}-(cos^(2)(x)+sin^()(x))*((∂)^2)/((∂)^( )x^2) [U[k](x)]-(e)^(x)[sum(1/(k[1]!){1/(k-k[1])(sum(sum(1/(k[3]!)*U[k[2]-k[3]](x)*U[k-k[1]-k[2]-1](x)},k[1]=0..k),k[2]=0..k-k[1]-1),k[3]=0..k[2])),U[k+1](x)];  od;"

Error, unable to match delimiters

"U[k+1](x):=solve((k+1)*U[k+1](x)+(x*(-1)^((k-1)/2))/(k!)-x^2*((e)^x)/10{6/(k!)-sum((2^(k[1]))/(k[1!])(5*delta[k-k[1]](x)+(2^(k-k[1])*(-1)^((k-k[1])/2))/((k-k[1])!)+(2^(k-k[1]+1)*(-1)^((k-k[1]-1)/2))/((k-k[1])!)),k[1]=0..k)}-(cos^2(x)+sin(x))*((∂)^2)/(∂x^2) [U[k](x)]-(e)^x[sum(1/(k[1]!){1/(k-k[1])(sum(sum(1/(k[3]!)*U[k[2]-k[3]](x)*U[k-k[1]-k[2]-1](x)},k[1]=0..k),k[2]=0..k-k[1]-1),k[3]=0..k[2])),U[k+1](x)];  od;"

 

``


 

Download Chapter_6-Example-6.5.4.mw

How to make the matrix construction by points animated?
I sketched this code - it doesn't work(((
 

a := Matrix([stoimostyM, izmeniya]);
plots[animate](plot, [a[n, m]], n = 1 .. 2970, m = 1 .. 2, numpoints = 50, frames = 100);
Error, bad index into Matrix

Hi! Do you know maybe how to solve equation with Laplace operator in Maple like BZ equation?

restart;
a := 0.75;
rho := u(t) + v(t) + w(t);
ode := diff(u(t), t) = 10*Delta(u(t)) + u*(-a*v - rho + 1);
ode1 := diff(v(t), t) = 10*Delta(v(t)) + v*(-a*w - rho + 1);
ode2 := diff(w(t), t) = 10*Delta(w(t)) + w*(-a*u - rho + 1);

 

Edit: Sorry I guess it should be function of three variables so u,v,w depends on (x,y,z) not strictle from time

I am wondering how to animate something like this from BZ equation:

Given a list L=[x,y,z], what is the best way to generate the following sequential expression?
L1=[[x],[y],[z],[x,x],[x,y],[[x,z],[y,y],[y,z],[z,z],[x,x,x],[x,x,y],[x,x,z],[x,y,y],[x,y,z],[y,y,y],[y,y,z],[y,z,z],[z,z,z]].

With Lin mind (a list of list of at most three variables), how can one generate a list of lists of any number of variable from L.

I am currently out of options, a response will be highly appreciated.

Thanks

 

I have Maple 2021 and Mapleflow 2021 installed. The Maple 2021 is working fine. However, MapleFlow gives the error as follows. ERROR: Flow does not evaluate to a module. My system OS is Windows 10 updates till April 2022

 

 

Details about our approach for phase-field models are provided in a paper just submitted. 
https://ecsarxiv.org/k2vu6/
 

Maple code (based on UMFPACK linearsolver and different compiled procedures for marching in time) is given here. Run the code for small values of NN and MM and keep increasing them. The goal is to do 100, 200, 400, etc and still get the code to run very fast.
 

restart:

t11:=time():

t12:=time[real]():

Digits:=15;

15

(1)

NN is the number of node points (elements) in the X and MM is the number of elements in the Y direction. delta is the applied current density at the top (Y =1). tf is the final time for simulation. vel is the velocity constant v in the paper. ki0 is the scaled exchange current density k in the paper. This code can be run for positive values of delta. This simulates plating. At the end of simulation, changing delta to negative values and rerunning the code will automatically used the geometry at the end of plating.
Ydatstore stores the geometry at every point in time. Phiaveadd stores the total liquid phase in the domain at any point in time.
Users can change NN, delta, tf, vel, ki0, MM just in this line and choose Edit execute worksheet to run for different design parameters.

Users can modify the call for y0proc for choosing different models.

Users can modify the call for HF to run first-order upwind, ENO2 or WENO3 methods. NN and MM should be even numbers.

NN:=100;delta:=0.1;tf:=2.0;vel:=1.0;ki0:=1.0;MM:=NN;

100

 

.1

 

2.0

 

1.0

 

1.0

 

100

(2)

N:=NN+2:h:=1.0/NN:M:=MM+2;Ny:=MM;

102

 

100

(3)

 

Initial geometry, Model 1, semicircle in a square

y0proc1:=proc(NN,MM,Y00)
local i,j,xx,yy,N,h,w,M,Ny,rf,f,ff,rr;

N:=NN+2:h:=1.0/NN:w:=h/2:

M:=MM+2;Ny:=MM;

for i from 2 to N-1 do for j from 2 to M-1 do
xx:=-0+(i-1/2-1)*h:yy:=-0+(j-1/2-1)*h:
rr:=xx^2+yy^2;
Y00[i,j]:=max(1e-9,0.5+0.5*tanh((sqrt(rr)-0.3)/w/sqrt(2.0))):
od:od:
for i from 1 to N do
Y00[i,1]:=Y00[i,2]:
Y00[i,M]:=Y00[i,M-1]:
od:
for j from 1 to M do
Y00[1,j]:=Y00[2,j]:
Y00[N,j]:=Y00[N-1,j]:
od:
end proc:

 

Model 2, square in a square

y0proc2:=proc(NN,MM,Y00)# square inside a circle, Model 2
local i,j,xx,yy,N,h,w,M,Ny,rf,f,ff,rr;

N:=NN+2:h:=1.0/NN:w:=h/2:

M:=MM+2;Ny:=MM;

for i from 2 to N-1 do for j from 2 to M-1 do
xx:=-0+(i-1/2-1)*h:yy:=-0+(j-1/2-1)*h:
if xx <=0.3 and yy<=0.3 then Y00[i,j]:=1e-9; else Y00[i,j]:=1.0:end:
od:od:
for i from 1 to N do
Y00[i,1]:=Y00[i,2]:
Y00[i,M]:=Y00[i,M-1]:
od:
for j from 1 to M do
Y00[1,j]:=Y00[2,j]:
Y00[N,j]:=Y00[N-1,j]:
od:
end proc:

 

 

Initial geometry, Model 3, electrodeposition problem trenches and via

y0proc3:=proc(NN,MM,Y00)
local i,j,xx,yy,N,h,w,M,Ny,rf,f,ff,rr;

N:=NN+2:h:=1.0/NN:w:=h/2:

M:=MM+2;Ny:=MM;

for i from 2 to N-1 do for j from 2 to M-1 do
xx:=-0+(i-1/2-1)*h:yy:=-0+(j-1/2-1)*h:
if abs(xx-0.5) >0.2 and yy<=0.5 then Y00[i,j]:=1e-9; else Y00[i,j]:=1.0:end:
od:od:
for i from 1 to N do
Y00[i,1]:=Y00[i,2]:
Y00[i,M]:=Y00[i,M-1]:
od:
for j from 1 to M do
Y00[1,j]:=Y00[2,j]:
Y00[N,j]:=Y00[N-1,j]:
od:
end proc:

 

 

Initial geometry, Model 4, Gaussian Seed at the bottom

y0proc4:=proc(NN,MM,Y00)
local i,j,xx,yy,N,h,w,M,Ny,rf,f,ff,rr;

N:=NN+2:h:=1.0/NN:w:=h/2:

M:=MM+2;Ny:=MM;

for i from 2 to N-1 do for j from 2 to M-1 do
xx:=-0+(i-1/2-1)*h:yy:=-0+(j-1/2-1)*h:
rr:=0.1+0.1*exp(-500.*(xx-0.5)^2);
Y00[i,j]:=max(1e-9,0.5+0.5*tanh((yy-rr)/w/sqrt(2.0))):
od:od:
for i from 1 to N do
Y00[i,1]:=Y00[i,2]:
Y00[i,M]:=Y00[i,M-1]:
od:
for j from 1 to M do
Y00[1,j]:=Y00[2,j]:
Y00[N,j]:=Y00[N-1,j]:
od:
end proc:

 

y0proc:=eval(y0proc1):#choose different models using y0proc2, etc.

Y00:=Matrix(1..N,1..M,datatype=float[8]):

evalhf(y0proc(NN,MM,Y00)):

if delta<0 then read("Y0data.m"):end:

if delta<0 then read("tdata.m"):end:

tf;

2.0

(4)

p0:=plots:-surfdata(Y00,-h/2..NN*h+h/2,-h/2..MM*h+h/2,dimension=2,style=surface,colorscheme = ["Red", "Green", "Blue"]):

printlevel:=2;

2

(5)

eq:=Array(1..N,1..M):

Nx:=NN:

 

 

Next, boundary conditiosn at X  = 0, X =1, Y = 0, Y = 1 are specified below, but these equations are not used and optimally coded inside the procedure Eqs11.

for i from 1 to 1 do for j from 1 to M do eq[i,j]:=-Phi[i,j]+Phi[i+1,j]:od:od:

for i from N to N do for j from 1 to M do eq[i,j]:=Phi[i-1,j]-Phi[i,j]:od:od:

for i from 1 to N do for j from 1 to 1 do eq[i,j]:=Phi[i,j+1]-Phi[i,j]:od:od:

for i from 1 to N do for j from M to M do eq[i,j]:=Phi[i,j-1]-Phi[i,j]+delta*h:od:od:

Eqs:=Vector(N*(M)):

N,M;

102, 102

(6)

 

Residues at different points in X and Y are coded in the Eqs11 procedure. Y0 is the input phase-field parameter (2D Matrix). The potential y is a vector to expedite the calculation of residue.

Eqs11:=proc(Y0::Matrix(datatype=float[8]),y::Vector(datatype=float[8]),delta::float,ki0::float,ff::Vector(datatype=float[8]))
local i::integer,j::integer,i1::integer,h::float[8];
global N,M;
h:=1.0/(N-2):

for i from 2 to N-1 do for j from 2 to M-1 do
i1:=i+(j-1)*N:
ff[i1]:=
(Y0[i,j]+Y0[i,j+1])*(y[i1+N]-y[i1])
-(Y0[i,j]+Y0[i,j-1])*(y[i1]-y[i1-N])
+(Y0[i+1,j]+Y0[i,j])*(y[i1+1]-y[i1])
-(Y0[i,j]+Y0[i-1,j])*(y[i1]-y[i1-1])
-ki0*y[i1]*h*(1e-24+(Y0[i+1,j]-Y0[i-1,j])^2+(Y0[i,j+1]-Y0[i,j-1])^2)^(1/2):
od:od:
for i from 1 to 1 do for j from 1 to M do
i1:=i+(j-1)*N:
ff[i1]:=-y[i1]+y[i1+1]:
od:od:
for i from N to N do for j from 1 to M do
i1:=i+(j-1)*N:
ff[i1]:=-y[i1]+y[i1-1]:
od:od:
for i from 1 to N do for j from 1 to 1 do
i1:=i+(j-1)*N:
ff[i1]:=-y[i1]+y[i1+N]:
od:od:
for i from 1 to N do for j from M to M do
i1:=i+(j-1)*N:
ff[i1]:=-y[i1]+y[i1-N]+delta*h:
od:od:
#print(1);
#ff;
end proc:

 

The Jacobian for the residues is coded int he procedure Jac1.

Jac1:=proc(Y0::Matrix(datatype=float[8]),y::Vector(datatype=float[8]),delta::float,ki0::float,j00::Matrix(datatype=float[8]))
local i::integer,j::integer,i1::integer,h::float[8];
global N,M;
h:=1.0/(N-2):

for i from 2 to N-1 do for j from 2 to M-1 do
i1:=i+(j-1)*N:
j00[i1,i1]:=-4*Y0[i,j]-Y0[i,j+1]-Y0[i,j-1]-Y0[i+1,j]-Y0[i-1,j]-ki0*h*(1e-24+(Y0[i+1,j]-Y0[i-1,j])^2+(Y0[i,j+1]-Y0[i,j-1])^2)^(1/2):
j00[i1,i1+1]:=Y0[i+1,j]+Y0[i,j]:j00[i1,i1-1]:=Y0[i-1,j]+Y0[i,j]:
j00[i1,i1+N]:=Y0[i,j+1]+Y0[i,j]:j00[i1,i1-N]:=Y0[i,j-1]+Y0[i,j]:
od:od:
for i from 1 to 1 do for j from 1 to M do
i1:=i+(j-1)*N:
j00[i1,i1]:=-1.0:j00[i1,i1+1]:=1.00:
#ff[i1]:=-y[i1]+y[i1+1]:
od:od:
for i from N to N do for j from 1 to M do
i1:=i+(j-1)*N:
j00[i1,i1]:=-1.0:j00[i1,i1-1]:=1.00:
#ff[i1]:=-y[i1]+y[i1-1]:
od:od:
for i from 1 to N do for j from 1 to 1 do
i1:=i+(j-1)*N:
#ff[i1]:=-y[i1]+y[i1+N]:
j00[i1,i1]:=-1.0:j00[i1,i1+N]:=1.00:
od:od:
for i from 1 to N do for j from M to M do
i1:=i+(j-1)*N:
j00[i1,i1]:=-1.0:j00[i1,i1-N]:=1.00:
#ff[i1]:=-y[i1]+y[i1-N]+delta*h:
od:od:
#print(1);
#ff;
end proc:

Eqs11:=Compiler:-Compile(Eqs11):

 

ntot:=N*(M);

10404

(7)

Ntot:=ntot;

10404

(8)

 

First-order Upwind method is coded in the procedure UpW.

UpW:=proc(Y00::Matrix(datatype=float[8]),Phi0::Vector(datatype=float[8]),F0::Matrix(datatype=float[8]),dt::float,N::integer,M::integer,v0::float)
local i::integer,j::integer,h::float[8],nx::float[8],vel::float[8],vx::float[8],vy::float[8],phix::float[8],phiy::float[8],phiave::float[8],jj::integer,phixb::float[8],phixf::float[8],phixb2::float[8],phixf2::float[8],vxb::float[8],phiyb::float[8],phiyf::float[8],vxf::float[8],phiyb2::float[8],phiyf2::float[8],vyb::float[8],vyf::float[8],uf::float[8],ub::float[8],vf::float[8],vb::float[8],tt::float[8],vv0::float[8],sd::float[8],sdf::float[8],sdb::float[8],sdx::float[8],sdy::float[8],sdxb::float[8],s1x::float[8],sdxf::float[8],sdyb::float[8],sdyf::float[8],s1y::float[8],vx1::float[8],vx2::float[8],vy1::float[8],vy2::float[8],w1::float[8],w2::float[8],r1::float[8],r2::float[8],alpha::float[8],beta::float[8];
#e1:=1e-15:

h:=1.0/(N-2):

vv0:=v0:
for i from 1 to N do
Y00[i,1]:=Y00[i,2]:
Y00[i,M]:=Y00[i,M-1]:
od:
for j from 1 to M do
Y00[1,j]:=Y00[2,j]:
Y00[N,j]:=Y00[N-1,j]:
od:
for i from 2 to N-1 do for j from 2 to M-1 do
vx:=0.0:vy:=0.0:phix:=0.0:phiy:=0.0:


vx1:=(Y00[i,j]-Y00[i-1,j])/h:
vx2:=(Y00[i+1,j]-Y00[i,j])/h:


vy1:=(Y00[i,j]-Y00[i,j-1])/h:
vy2:=(Y00[i,j+1]-Y00[i,j])/h:

if v0>=0 then
vx1:=max(vx1,0):vx2:=-min(vx2,0): else
vx1:=-min(vx1,0):vx2:=max(vx2,0): end:

if v0>=0 then
vy1:=max(vy1,0):vy2:=-min(vy2,0): else
vy1:=-min(vy1,0):vy2:=max(vy2,0): end:
nx:=sqrt(max(vx1,vx2)^2+max(vy1,vy2)^2):
F0[i,j]:=nx:
od:od:

end proc:

 

Second-order ENO2 method is coded in the procedure ENO2.

ENO2:=proc(Y00::Matrix(datatype=float[8]),Phi0::Vector(datatype=float[8]),F0::Matrix(datatype=float[8]),dt::float,N::integer,M::integer,v0::float)
local i::integer,j::integer,h::float[8],nx::float[8],vel::float[8],vx::float[8],vy::float[8],phix::float[8],phiy::float[8],phiave::float[8],jj::integer,phixb::float[8],phixf::float[8],phixb2::float[8],phixf2::float[8],vxb::float[8],phiyb::float[8],phiyf::float[8],vxf::float[8],phiyb2::float[8],phiyf2::float[8],vyb::float[8],vyf::float[8],uf::float[8],ub::float[8],vf::float[8],vb::float[8],tt::float[8],vv0::float[8],sd::float[8],sdf::float[8],sdb::float[8],sdx::float[8],sdy::float[8],sdxb::float[8],s1x::float[8],sdxf::float[8],sdyb::float[8],sdyf::float[8],s1y::float[8],vx1::float[8],vx2::float[8],vy1::float[8],vy2::float[8],alpha::float[8],beta::float[8];


h:=1.0/(N-2):
for i from 1 to N do
Y00[i,1]:=Y00[i,2]:
Y00[i,M]:=Y00[i,M-1]:
od:
for j from 1 to M do
Y00[1,j]:=Y00[2,j]:
Y00[N,j]:=Y00[N-1,j]:
od:
for i from 2 to N-1 do for j from 2 to M-1 do
vx:=0.0:vy:=0.0:phix:=0.0:phiy:=0.0:

sdx:=(Y00[i+1,j]-2*Y00[i,j]+Y00[i-1,j])/h:sdy:=(Y00[i,j+1]-2*Y00[i,j]+Y00[i,j-1])/h:
vxb:=0:vxf:=0:vyb:=0:vyf:=0:
if i = 2 then sdxb:=(Y00[i,j]-2*Y00[i-1,j]+Y00[i-1,j])/h:else sdxb:=(Y00[i,j]-2*Y00[i-1,j]+Y00[i-2,j])/h:end:
if sdx*sdxb>=0 then s1x:=1.0 else s1x:=0.0:end:
vx1:=(Y00[i,j]-Y00[i-1,j])/h+0.5*signum(sdx)*s1x*min(abs(sdx),abs(sdxb)):

if i = N-1 then sdxf:=(Y00[i+1,j]-2*Y00[i+1,j]+Y00[i,j])/h:else sdxf:=(Y00[i+2,j]-2*Y00[i+1,j]+Y00[i,j])/h:end:
if sdx*sdxf>=0 then s1x:=1.0 else s1x:=0.0:end:
vx2:=(Y00[i+1,j]-Y00[i,j])/h-0.5*signum(sdx)*s1x*min(abs(sdx),abs(sdxf)):


if j = 2 then sdyb:=(Y00[i,j]-2*Y00[i,j-1]+Y00[i,j-1])/h:else sdyb:=(Y00[i,j]-2*Y00[i,j-1]+Y00[i,j-2])/h:end:
if sdy*sdyb>=0 then s1y:=1.0 else s1y:=0.0:end:
vy1:=(Y00[i,j]-Y00[i,j-1])/h+0.5*signum(sdy)*s1y*min(abs(sdy),abs(sdyb)):

if j = M-1 then sdyf:=(Y00[i,j+1]-2*Y00[i,j+1]+Y00[i,j])/h:else sdyf:=(Y00[i,j+2]-2*Y00[i,j+1]+Y00[i,j])/h:end:
if sdy*sdyf>=0 then s1y:=1.0 else s1y:=0.0:end:
vy2:=(Y00[i,j+1]-Y00[i,j])/h+0.5*signum(sdy)*s1y*min(abs(sdy),abs(sdyf)):

if v0>=0 then
vx1:=max(vx1,0):vx2:=-min(vx2,0): else
vx1:=-min(vx1,0):vx2:=max(vx2,0): end:

if v0>=0 then
vy1:=max(vy1,0):vy2:=-min(vy2,0): else
vy1:=-min(vy1,0):vy2:=max(vy2,0): end:
nx:=sqrt(max(vx1,vx2)^2+max(vy1,vy2)^2):

F0[i,j]:=nx:
od:od:

end proc:

 

Third-order WENO3 method is coded in the procedure WENO3.

WENO3:=proc(Y00::Matrix(datatype=float[8]),Phi0::Vector(datatype=float[8]),F0::Matrix(datatype=float[8]),dt::float,N::integer,M::integer,v0::float)
local i::integer,j::integer,h::float[8],nx::float[8],vel::float[8],vx::float[8],vy::float[8],phix::float[8],phiy::float[8],phiave::float[8],jj::integer,phixb::float[8],phixf::float[8],phixb2::float[8],phixf2::float[8],vxb::float[8],phiyb::float[8],phiyf::float[8],vxf::float[8],phiyb2::float[8],phiyf2::float[8],vyb::float[8],vyf::float[8],uf::float[8],ub::float[8],vf::float[8],vb::float[8],tt::float[8],vv0::float[8],sd::float[8],sdf::float[8],sdb::float[8],sdx::float[8],sdy::float[8],sdxb::float[8],s1x::float[8],sdxf::float[8],sdyb::float[8],sdyf::float[8],s1y::float[8],vx1::float[8],vx2::float[8],vy1::float[8],vy2::float[8],w1::float[8],w2::float[8],r1::float[8],r2::float[8],alpha::float[8],beta::float[8],e1::float[8];
e1:=1e-6:

h:=1.0/(N-2):

vv0:=v0:
for i from 1 to N do
Y00[i,1]:=Y00[i,2]:
Y00[i,M]:=Y00[i,M-1]:
od:
for j from 1 to M do
Y00[1,j]:=Y00[2,j]:
Y00[N,j]:=Y00[N-1,j]:
od:
for i from 2 to N-1 do for j from 2 to M-1 do
vx:=0.0:vy:=0.0:phix:=0.0:phiy:=0.0:
phix:=(Y00[i+1,j]-Y00[i-1,j])/2/h:
phiy:=(Y00[i,j+1]-Y00[i,j-1])/2/h:

if i = 2 then sdb:=Y00[i,j]-2*Y00[i-1,j]+Y00[i-1,j]: else sdb:=Y00[i,j]-2*Y00[i-1,j]+Y00[i-2,j]:end:
sd:=Y00[i+1,j]-2*Y00[i,j]+Y00[i-1,j]:
if i = N-1 then sdf:=Y00[i,j]-2*Y00[i+1,j]+Y00[i+1,j]: else sdf:=Y00[i+2,j]-2*Y00[i+1,j]+Y00[i,j]:end:
r1:=(e1+sdb^2)/(e1+sd^2):w1:=1/(1+2*r1^2):r2:=(e1+sdf^2)/(e1+sd^2):w2:=1/(1+2*r2^2):
vx1:=phix-0.5*w1/h*(sd-sdb):
vx2:=phix-0.5*w2/h*(sdf-sd):

if j = 2 then sdb:=Y00[i,j]-2*Y00[i,j-1]+Y00[i,j-1]:else sdb:=Y00[i,j]-2*Y00[i,j-1]+Y00[i,j-2]:end:
sd:=Y00[i,j+1]-2*Y00[i,j]+Y00[i,j-1]:
if j = M-1 then sdf:=Y00[i,j]-2*Y00[i,j+1]+Y00[i,j+1]: else sdf:=Y00[i,j+2]-2*Y00[i,j+1]+Y00[i,j]:end:
r1:=(e1+sdb^2)/(e1+sd^2):w1:=1/(1+2*r1^2):r2:=(e1+sdf^2)/(e1+sd^2):w2:=1/(1+2*r2^2):

vy1:=phiy-0.5*w1/h*(sd-sdb):
vy2:=phiy-0.5*w2/h*(sdf-sd):

if v0>=0 then
vx1:=max(vx1,0):vx2:=-min(vx2,0): else
vx1:=-min(vx1,0):vx2:=max(vx2,0): end:

if v0>=0 then
vy1:=max(vy1,0):vy2:=-min(vy2,0): else
vy1:=-min(vy1,0):vy2:=max(vy2,0): end:
nx:=sqrt(max(vx1,vx2)^2+max(vy1,vy2)^2):
#nx:=sqrt(vx1^2+vx2^2+vy1^2+vy^2):

F0[i,j]:=nx:
od:od:

end proc:

 

UpW:=Compiler:-Compile(UpW):

ENO2:=Compiler:-Compile(ENO2):

WENO3:=Compiler:-Compile(WENO3):

F0:=Matrix(1..N,1..M,datatype=float[8]):

h/0.1/2;

0.500000000000000e-1

(9)

 

Nx;

100

(10)

PhiAdd:=proc(N::integer,Phi0::Vector(datatype=float[8]),db::Vector(datatype=float[8]))
local i::integer;
for i from 1 to N do Phi0[i]:=Phi0[i]+db[i]:od:
end proc:

PhiAdd:=Compiler:-Compile(PhiAdd):

Ntot:=ntot:

Phi0:=Vector(1..Ntot,datatype=float[8]):

printlevel:=1:

ff:=copy(Phi0):

 

Phi0:=Vector(1..Ntot,datatype=float[8]):

j00:=Matrix(1..Ntot,1..Ntot,datatype=float[8],storage=sparse):

evalf(Eqs11(Y00,Phi0,delta,ki0,ff));

0.100000000000000e-2

(11)

Jac1(Y00,Phi0,delta,ki0,j00):

db:=LinearAlgebra:-LinearSolve(j00,-ff,method=SparseDirect):

Phi0:=Phi0+db:

V[0]:=(Phi0[ntot-2*N+N/2]/2+Phi0[ntot-2*N+N/2+1]/2)+h/2*delta;

HFloat(0.31581237885872887)

(12)

TT[0]:=0;tt:=0:

0

(13)

vv0:=max(abs(Phi0[ntot-2*N+1]),abs(Phi0[ntot-2*N+N/2]),abs(Phi0[ntot-2*N+N/2+1]),abs(Phi0[ntot-N])):

dt:=min(h/vv0/vel/ki0,tf-tt);

HFloat(0.031202770994846224)

(14)

Ymid:=Matrix(1..N,1..M,datatype=float[8]):

phiaveAdd:=proc(Y00::Matrix(datatype=float[8]),N::integer,M::integer,Ny::integer)
local i::integer,j::integer,phiave::float;
phiave:=0.0:
for i from 2 to N-1 do for j from 2 to M-1 do phiave:=phiave+Y00[i,j]:od:od:
phiave/(N-2)/(M-2);
end proc:

phiaveAdd:=Compiler:-Compile(phiaveAdd):

Ny;M;

100

 

102

(15)

phiaveAdd(Y00,N,M,Ny);

.929280193991299241

(16)

 

The three stages of  SSR-RK3 are stored in EFAdd, EFAdd2 and EFAdd3

EFAdd:=proc(Y00::Matrix(datatype=float[8]),Ymid::Matrix(datatype=float[8]),Phi0::Vector(datatype=float[8]),F0::Matrix(datatype=float[8]),dt::float[8],vel::float[8],ki0::float[8],N::integer,M::integer)
local i::integer,j::integer;
#for i from 2 to N-1 do for j from 2 to M-1 do Ymid[i,j]:=Y00[i,j]-dt*F0[i,j]*Phi0[i+(j-1)*N]:od:od:
for i from 2 to N-1 do for j from 2 to M-1 do Ymid[i,j]:=max(1e-9,Y00[i,j]-dt*vel*ki0*F0[i,j]*Phi0[i+(j-1)*N]):od:od:
for i from 1 to N do
Ymid[i,1]:=Ymid[i,2]:
Ymid[i,M]:=Ymid[i,M-1]:
od:
for j from 1 to M do
Ymid[1,j]:=Ymid[2,j]:
Ymid[N,j]:=Ymid[N-1,j]:
od:
end proc:

EFAdd:=Compiler:-Compile(EFAdd):

EFAdd2:=proc(Y00::Matrix(datatype=float[8]),Ymid::Matrix(datatype=float[8]),Phi0::Vector(datatype=float[8]),F0::Matrix(datatype=float[8]),dt::float[8],vel::float[8],ki0::float[8],N::integer,M::integer)
local i::integer,j::integer;
for i from 2 to N-1 do for j from 2 to M-1 do
Ymid[i,j]:=max(1e-9,Y00[i,j]*3/4.+Ymid[i,j]/4.-dt/4.*vel*ki0*F0[i,j]*Phi0[i+(j-1)*N]):od:od:
for i from 1 to N do
Ymid[i,1]:=Ymid[i,2]:
Ymid[i,M]:=Ymid[i,M-1]:
od:
for j from 1 to M do
Ymid[1,j]:=Ymid[2,j]:
Ymid[N,j]:=Ymid[N-1,j]:
od:
end proc:

EFAdd2:=Compiler:-Compile(EFAdd2):

EFAdd3:=proc(Y00::Matrix(datatype=float[8]),Ymid::Matrix(datatype=float[8]),Phi0::Vector(datatype=float[8]),F0::Matrix(datatype=float[8]),dt::float[8],vel::float[8],ki0::float[8],N::integer,M::integer)
local i::integer,j::integer;
for i from 2 to N-1 do for j from 2 to M-1 do
Y00[i,j]:=max(1e-9,Y00[i,j]*1/3.+Ymid[i,j]*2/3.-dt*2/3.*vel*ki0*F0[i,j]*Phi0[i+(j-1)*N]):od:od:
for i from 1 to N do
Y00[i,1]:=Y00[i,2]:
Y00[i,M]:=Y00[i,M-1]:
od:
for j from 1 to M do
Y00[1,j]:=Y00[2,j]:
Y00[N,j]:=Y00[N-1,j]:
od:
end proc:

EFAdd3:=Compiler:-Compile(EFAdd3):

 

YdatStore:=proc(Y00::Matrix(datatype=float[8]),Ydat::Array(datatype=float[8]),N::integer,M::integer,jj::integer)
local i::integer,j::integer;
for i from 2 to N-1 do for j from 2 to M-1 do
Ydat[jj,i,j]:=Y00[i,j]:od:od:
end proc:

YdatStore:=Compiler:-Compile(YdatStore):

Phiave[0]:=phiaveAdd(Y00,N,M,Ny);

.929280193991299241

(17)

 

Nt:=round(tf/dt)+50;

114

(18)

printlevel:=1:

Ymid:=copy(Y00):

Ydat:=Array(1..Nt+1,1..N,1..M,datatype=float[8]):

YdatStore(Y00,Ydat,N,M,1);

1.

(19)

 

ii:=0:TT[0];tt;

0

 

0

(20)

 

Different upwind schemes can be called by assign WENO3, UpW or ENO3 scheme.

HF:=eval(WENO3):

#HF:=eval(UpW):

#HF:=eval(ENO2):

 

A while loop is written from t=0 to t= tf.

while tt<tf do
HF(Y00,Phi0,F0,evalf(dt),N,M,delta):
EFAdd(Y00,Ymid,Phi0,F0,dt,vel,ki0,N,M);
evalf(Eqs11(Ymid,Phi0,delta,ki0,ff));
Jac1(Ymid,Phi0,delta,ki0,j00):
db:=LinearAlgebra:-LinearSolve(j00,-ff,method=SparseDirect):
PhiAdd(Ntot,Phi0,db):
HF(Ymid,Phi0,F0,evalf(dt),N,M,delta):
EFAdd2(Y00,Ymid,Phi0,F0,dt,vel,ki0,N,M);
evalf(Eqs11(Ymid,Phi0,delta,ki0,ff));
Jac1(Ymid,Phi0,delta,ki0,j00):
db:=LinearAlgebra:-LinearSolve(j00,-ff,method=SparseDirect):
PhiAdd(Ntot,Phi0,db):
HF(Ymid,Phi0,F0,evalf(dt),N,M,delta):
EFAdd3(Y00,Ymid,Phi0,F0,dt,vel,ki0,N,M);
evalf(Eqs11(Y00,Phi0,delta,ki0,ff));
Jac1(Y00,Phi0,delta,ki0,j00):
db:=LinearAlgebra:-LinearSolve(j00,-ff,method=SparseDirect):
PhiAdd(Ntot,Phi0,db):
ii:=ii+1:#print(i);
V[ii]:=(Phi0[ntot-2*N+N/2]/2+Phi0[ntot-2*N+N/2+1]/2)+h/2*delta;#print(ii,V[ii]);
TT[ii]:=TT[ii-1]+dt:tt:=tt+dt:
YdatStore(Y00,Ydat,N,M,ii+1);
Phiave[ii]:=phiaveAdd(Y00,N,M,Ny);
vv0:=max(abs(Phi0[ntot-2*N+1]),abs(Phi0[ntot-2*N+N/2]),abs(Phi0[ntot-2*N+N/2+1]),abs(Phi0[ntot-N])):
dt:=min(h/vv0/vel/ki0,tf-tt);#print(tt,ii,dt);
#YdatStore(Y00,Ydat,N,M,i+1);
#Phiave[i]:=phiaveAdd(Y00,N,M,Ny);
end:

 

nt:=ii;

46

(21)

V[nt];

HFloat(0.17021428328406052)

(22)

 

Voltage time curves are plotted below. Voltage is measured at X = 0.5, Y = 1.

plot([[seq([TT[ii],V[ii]],ii=0..nt)]],style=point);

 

 

Voltage at the end of plating, cpu time can be documented as

V[nt];time[real]()-t12;time()-t11;NN;

HFloat(0.17021428328406052)

 

19.042

 

14.719

 

100

(23)

 

p1:=plots:-surfdata(Y00,-h/2..NN*h+h/2,-h/2..MM*h+h/2,dimension=2,style=surface,colorscheme = ["Red", "Green", "Blue"]):

tf:=TT[nt];

HFloat(2.0)

(24)

Contour plots at t= 0 and t = 2.0 (at the of plating are given below)

plots:-display({p0});plots:-display({p1});

 

 

 

 

The liquid phase content as a funciton of time is plotted below

plot([[seq([TT[ii],Phiave[ii]],ii=0..nt)]],style=point);

 

 

save Y00,"Y0data.m";

save tf,"tdata.m";

read("Y0data.m"):

plots:-surfdata(Y00,-h/2..NN*h+h/2,-h/2..MM*h+h/2,dimension=2,style=surface,colorscheme = ["Red", "Green", "Blue"]);

 

 


 

Download Phasefield2DBaseCodeParametric.mw
 

restart:

t11:=time():

t12:=time[real]():

Digits:=15;

15

(1)

NN is the number of node points (elements) in the X and MM is the number of elements in the Y direction. delta is the applied current density at the top (Y =1). tf is the final time for simulation. vel is the velocity constant v in the paper. ki0 is the scaled exchange current density k in the paper. This code can be run for positive values of delta. This simulates plating. At the end of simulation, changing delta to negative values and rerunning the code will automatically used the geometry at the end of plating.
Ydatstore stores the geometry at every point in time. Phiaveadd stores the total liquid phase in the domain at any point in time.
Users can change NN, delta, tf, vel, ki0, MM just in this line and choose Edit execute worksheet to run for different design parameters.

Users can modify the call for y0proc for choosing different models.

Users can modify the call for HF to run first-order upwind, ENO2 or WENO3 methods. NN and MM should be even numbers.

NN:=100;delta:=0.1;tf:=2.0;vel:=1.0;ki0:=1.0;MM:=NN;

100

 

.1

 

2.0

 

1.0

 

1.0

 

100

(2)

N:=NN+2:h:=1.0/NN:M:=MM+2;Ny:=MM;

102

 

100

(3)

 

Initial geometry, Model 1, semicircle in a square

y0proc1:=proc(NN,MM,Y00)
local i,j,xx,yy,N,h,w,M,Ny,rf,f,ff,rr;

N:=NN+2:h:=1.0/NN:w:=h/2:

M:=MM+2;Ny:=MM;

for i from 2 to N-1 do for j from 2 to M-1 do
xx:=-0+(i-1/2-1)*h:yy:=-0+(j-1/2-1)*h:
rr:=xx^2+yy^2;
Y00[i,j]:=max(1e-9,0.5+0.5*tanh((sqrt(rr)-0.3)/w/sqrt(2.0))):
od:od:
for i from 1 to N do
Y00[i,1]:=Y00[i,2]:
Y00[i,M]:=Y00[i,M-1]:
od:
for j from 1 to M do
Y00[1,j]:=Y00[2,j]:
Y00[N,j]:=Y00[N-1,j]:
od:
end proc:

 

Model 2, square in a square

y0proc2:=proc(NN,MM,Y00)# square inside a circle, Model 2
local i,j,xx,yy,N,h,w,M,Ny,rf,f,ff,rr;

N:=NN+2:h:=1.0/NN:w:=h/2:

M:=MM+2;Ny:=MM;

for i from 2 to N-1 do for j from 2 to M-1 do
xx:=-0+(i-1/2-1)*h:yy:=-0+(j-1/2-1)*h:
if xx <=0.3 and yy<=0.3 then Y00[i,j]:=1e-9; else Y00[i,j]:=1.0:end:
od:od:
for i from 1 to N do
Y00[i,1]:=Y00[i,2]:
Y00[i,M]:=Y00[i,M-1]:
od:
for j from 1 to M do
Y00[1,j]:=Y00[2,j]:
Y00[N,j]:=Y00[N-1,j]:
od:
end proc:

 

 

Initial geometry, Model 3, electrodeposition problem trenches and via

y0proc3:=proc(NN,MM,Y00)
local i,j,xx,yy,N,h,w,M,Ny,rf,f,ff,rr;

N:=NN+2:h:=1.0/NN:w:=h/2:

M:=MM+2;Ny:=MM;