Parallel Method for solving a Coupled PDE

Parallel Method for solving a

Coupled PDE and its application in simulation of thin composite films

Parallel Method for solving a

Coupled PDE and its application in simulation of thin composite films

By: E. NedaaeeIASBS

What is thin film

is thermal fluctuations of the flux of the incoming particles, such that

represents the fluctuations in the density of the incoming particles with

, the order parameter which represents the difference between the densities of the two types of particles of the surface , the height of the surface

II ),()(2

.)(

I ),(2

)(2

2232

222

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)()(2),().,( ttXXDtXtx dmmm

)()(2),().,( ttXXDtXtx dhhh

),( tXh

),( tXm

h

m

Our Model:

0)()(8

)4(

))(())((4

)()4(

0)(2

)()(8

)4(

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Discretization

Fully implicit finite difference method

nj

ni

i

j

ni

i

j

ni

i

:

1

2:

1

1:

1

m

h

Start

Initial Conditions

maxtt

Solve Nonlinear Set of Equations.

Finish

mm

hh

uuold

uuold

ttt

Yes

No

Algorithm

Nonlinear SolverNewton-Raphson Method

0),,,( 21 ni xxxF

)()()( 2

1

XOxx

FXFXXF j

n

j j

iii

j

iij x

FJ

)(.)()( 2XOXJXFXXF

)(. XFXJ

Initial guess for X

Solve.

)(. XFXJ

Set.XXX oldnew

X

FNo Yes Return newX

Solving this linear set of equations by Conjugate-Gradient method and

store J with sparse manner

Parallelization MethodDomain Decomposition

maxtt

Solve Nonlinear Set of Equations. In each domain

mm

hh

uut

uut

Yes

No

z

Solve Nonlinear Set of Equations. In each domain uut

Finish

mm

hh

uuold

uuold

ttt

No

Yes

B.C. Exchange

MPI_Init(&argc, &argv);MPI_Comm_size(MPI_COMM_WORLD, &nump);MPI_Comm_rank(MPI_COMM_WORLD, &myrank);nump_row = (int) sqrt(nump);dims[0] = nump_row; dims[1] = nump_row;periods[0] = periods[1] = 1;

MPI_Cart_create(MPI_COMM_WORLD, 2, dims, periods, 0, &comm);MPI_Comm_rank(comm, &myrank);MPI_Cart_coords(comm, myrank, 2, coords);MPI_Type_hvector(np,1,(np+2)*sizeof(double), MPI_DOUBLE, &BC);MPI_Type_commit(&BC);

rcoords[0] = coords[0]; rcoords[1] = coords[1]+1; if(coords[1] == (nump_row-1)) rcoords[1] = 0;

lcoords[0] = coords[0]; lcoords[1] = coords[1]-1; if(coords[1] == 0) lcoords[1] = nump_row - 1;

bcoords[1] = coords[1]; bcoords[0] = coords[0]+1; if(coords[0] == (nump_row-1)) bcoords[0] = 0;

tcoords[1] = coords[1]; tcoords[0] = coords[0]-1; if(coords[0] == 0) tcoords[0] = nump_row - 1;

MPI_Cart_rank(comm, lcoords, &left);MPI_Cart_rank(comm, rcoords, &right);MPI_Cart_rank(comm, tcoords, &top);MPI_Cart_rank(comm, bcoords, &buttom);

MPI_File_open(comm, fh_m_out,MPI_MODE_CREATE | MPI_MODE_RDWR, MPI_INFO_NULL, &fh_m);

MPI_File_open(comm, fh_h_out,MPI_MODE_CREATE | MPI_MODE_RDWR, MPI_INFO_NULL, &fh_h);

MPI_Barrier(comm);

lsizes[0] = np;lsizes[1] = np;

gsizes[0] = np*nump_row*file_size;gsizes[1] = np*nump_row;

memsizes[0] = lsizes[0] + 2;memsizes[1] = lsizes[1] + 2;

start_indices[0] = start_indices[1] = 1;

MPI_Type_create_subarray(2,memsizes, lsizes, start_indices, MPI_ORDER_C,MPI_DOUBLE, &memtype);

MPI_Type_commit(&memtype);

start_indices[0] = coords[0]*lsizes[0]+count_ind*np*nump_row; start_indices[1] = coords[1]*lsizes[1];

MPI_Type_create_subarray(2,gsizes,lsizes,start_indices,MPI_ORDER_C,MPI_DOUBLE,&filetype);

MPI_Type_commit(&filetype);

MPI_File_set_view(fh_m, 0, MPI_DOUBLE, filetype, "native", MPI_INFO_NULL);

MPI_File_set_view(fh_h, 0, MPI_DOUBLE, filetype, "native", MPI_INFO_NULL);

MPI_File_write_all(fh_m,u_m, 1, memtype, &status);MPI_Wait(&req, &status);MPI_File_write_all(fh_h,u_h, 1, memtype, &status);MPI_Wait(&req, &status);MPI_Barrier(comm);count_ind +=1;

for(;;){ vecfunc(x); newt(x, &check,vecfunc);

MPI_Barrier(comm);BC_exchange(left,right,top,buttom);MPI_Barrier(comm);pnorm_s = NORM(x,x_temp);

MPI_Allreduce(&pnorm_s, &pnorm_r, 1, MPI_DOUBLE, MPI_SUM, MPI_COMM_WORLD);

if(sqrt(pnorm_r) <= 1.e-9) break;for(int i = 1; i<= 2*np*np; i++)

x_temp[i] = x[i];count +=1;

}

void BC_exchange(int left, int right, int top, int buttom){

/*send and received from left neighbor*/ MPI_Isend(&u_h[1][1],np,MPI_DOUBLE, top,0,MPI_COMM_WORLD, &requests[0]); MPI_Irecv(&u_h[0][1],np,MPI_DOUBLE, top,0,MPI_COMM_WORLD, &requests[1]); /*send and received from right neighbor*/ MPI_Isend(&u_h[np][1],np,MPI_DOUBLE, buttom,0,MPI_COMM_WORLD, &requests[2]); MPI_Irecv(&u_h[np+1][1],np,MPI_DOUBLE,buttom,0,MPI_COMM_WORLD, &requests[3]); /*send and received from top neighbor*/ MPI_Isend(&u_h[1][np],1,BC, right,0,MPI_COMM_WORLD, &requests[4]); MPI_Irecv(&u_h[1][np+1],1,BC,right,0,MPI_COMM_WORLD, &requests[5]);

/*send and received from bottom neighbor*/ MPI_Isend(&u_h[1][1],1,BC, left,0,MPI_COMM_WORLD, &requests[6]); MPI_Irecv(&u_h[1][0],1,BC, left,0,MPI_COMM_WORLD, &requests[7]);

MPI_Waitall(8,requests, statuses);