[petsc-users] nonzero prescribed boundary condition

Thu Sep 8 02:02:35 CDT 2011

Hi,
Again about prescribed boundary condition issue.

>>>> So my quick follow up question is How a displacement-controled load is
>>>> done corrently in Petsc 2.3.3?
>>>
>>> To do it in 2.3.3 simply have for those components of F() the formula
>>> F_i = x_i - givenvalue_i and in your Jacobian just use MatZeroRows() for
>>> those rows
>>>

in a nonlinear solving, I followed the above suggestion: in FormFunction,
set F_i=x_i-given_value (nonzero) and in FormJacobian, MatZeroRows for the 
corresponding rows.
However, the convergence is must slowed down as the exact satisfaction of 
xi=givenvalue
has to be satisfied in an interative manner. Acturally convergence is not 
just slow but almost
not practical (normally > 50 interations).

Am I am missing something?

Rong

----- Original Message ----- 
From: "Tian(ICT)" <rongtian at ncic.ac.cn>
To: "PETSc users list" <petsc-users at mcs.anl.gov>
Sent: Wednesday, May 11, 2011 2:16 PM
Subject: Re: [petsc-users] nonzero prescribed boundary condition

> With additional -snes_type test:
>
> "-pc_type
> lu -ksp_monitor_true_residual -snes_monitor -ksp_monitor -snes_type test"
>
> atol=1e-050, rtol=1e-008, stol=1e-008, maxit=50, maxf=10000
> Testing hand-coded Jacobian, if the ratio is
> O(1.e-8), the hand-coded Jacobian is probably correct.
> Run with -snes_test_display to show difference
> of hand-coded and finite difference Jacobian.
> Norm of matrix ratio 9.05702e-008 difference 3.94971
> Norm of matrix ratio 3.6015e-007 difference 15.7059
> Norm of matrix ratio 3.73941e-007 difference 16.3073
> [0]PETSC ERROR: SNESSolve() line 1871 in
> src/snes/interface/C:\cygwin\home\PETSC-~1.3-P\src\snes\INTERF~1\snes.c
> [0]PETSC ERROR: User provided function() line 572 in
> unknowndirectory/c:\cygwin\home\rabbit\pgfem - csrbf\domain.cpp
> [0]PETSC ERROR: User provided function() line 40 in
> unknowndirectory/c:\cygwin\home\rabbit\pgfem - csrbf\main.cpp
>
>
> -snes_mf_operator does not work:
>
> "-pc_type
> lu -ksp_monitor_true_residual -snes_monitor -ksp_monitor -snes_ms_operator"
>
> [0]PETSC ERROR: --------------------- Error 
> Message ------------------------------------
> [0]PETSC ERROR: Object is in wrong state!
> [0]PETSC ERROR: Not for unassembled matrix!
> [0]PETSC 
> ERROR: ------------------------------------------------------------------------
> [0]PETSC ERROR: Petsc Release Version 2.3.3, Patch 15, Tue Sep 23 10:02:49 
> CDT 2008 HG revision: 31306062cd1a6f6a2496fccb4878f485c9b91760
> [0]PETSC ERROR: See docs/changes/index.html for recent updates.
> [0]PETSC ERROR: See docs/faq.html for hints about trouble shooting.
> [0]PETSC ERROR: See docs/index.html for manual pages.
> [0]PETSC 
> ERROR: ------------------------------------------------------------------------
> [0]PETSC ERROR: pgfem on a cygwin-gc named RTI458-LAPTOP by rabbit Wed May 
> 11 14:16:08 2011
> [0]PETSC ERROR: Libraries linked from /home/petsc-2.3.3-p15/lib/cygwin-gcc
> [0]PETSC ERROR: Configure run at Wed Dec  1 17:02:20 2010
> [0]PETSC ERROR: Configure options --with-cc="win32fe 
> cl" --with-cxx="win32fe 
> cl" --download-c-blas-lapack=1 --with-clanguage=cxx --with-fc=0 --useThreads=0 
>  --with-shared=0
> [0]PETSC 
> ERROR: ------------------------------------------------------------------------
> [0]PETSC ERROR: MatDuplicate() line 3218 in 
> src/mat/interface/C:\cygwin\home\PETSC-~1.3-P\src\mat\INTERF~1\matrix.c
> atol=1e-050, rtol=1e-008, stol=1e-008, maxit=50, maxf=10000
>  0 SNES Function norm 7.071067811865e-002
>  0 KSP Residual norm 9.965778978387e-002
> [0]PETSC ERROR: --------------------- Error 
> Message ------------------------------------
> [0]PETSC ERROR: Null argument, when expecting valid pointer!
> [0]PETSC ERROR: Null Object: Parameter # 1!
> [0]PETSC 
> ERROR: ------------------------------------------------------------------------
> [0]PETSC ERROR: Petsc Release Version 2.3.3, Patch 15, Tue Sep 23 10:02:49 
> CDT 2008 HG revision: 31306062cd1a6f6a2496fccb4878f485c9b91760
> [0]PETSC ERROR: See docs/changes/index.html for recent updates.
> [0]PETSC ERROR: See docs/faq.html for hints about trouble shooting.
> [0]PETSC ERROR: See docs/index.html for manual pages.
> [0]PETSC 
> ERROR: ------------------------------------------------------------------------
> [0]PETSC ERROR: pgfem on a cygwin-gc named RTI458-LAPTOP by rabbit Wed May 
> 11 14:16:08 2011
> [0]PETSC ERROR: Libraries linked from /home/petsc-2.3.3-p15/lib/cygwin-gcc
> [0]PETSC ERROR: Configure run at Wed Dec  1 17:02:20 2010
> [0]PETSC ERROR: Configure options --with-cc="win32fe 
> cl" --with-cxx="win32fe 
> cl" --download-c-blas-lapack=1 --with-clanguage=cxx --with-fc=0 --useThreads=0 
>  --with-shared=0
> [0]PETSC 
> ERROR: ------------------------------------------------------------------------
> [0]PETSC ERROR: PetscObjectGetComm() line 34 in 
> src/sys/objects/C:\cygwin\home\PETSC-~1.3-P\src\sys\objects\gcomm.c
> [0]PETSC ERROR: VecNormBegin() line 495 in 
> src/vec/vec/utils/C:\cygwin\home\PETSC-~1.3-P\src\vec\vec\utils\comb.c
> [0]PETSC ERROR: MatMFFDCompute_WP() line 73 in 
> src/mat/impls/mffd/C:\cygwin\home\PETSC-~1.3-P\src\mat\impls\mffd\wp.c
> [0]PETSC ERROR: MatMult_MFFD() line 294 in 
> src/mat/impls/mffd/C:\cygwin\home\PETSC-~1.3-P\src\mat\impls\mffd\mffd.c
> [0]PETSC ERROR: MatMult() line 1632 in 
> src/mat/interface/C:\cygwin\home\PETSC-~1.3-P\src\mat\INTERF~1\matrix.c
> [0]PETSC ERROR: KSPDefaultBuildResidual() line 605 in 
> src/ksp/ksp/interface/C:\cygwin\home\PETSC-~1.3-P\src\ksp\ksp\INTERF~1\iterativ.c
> [0]PETSC ERROR: KSPBuildResidual() line 1587 in 
> src/ksp/ksp/interface/C:\cygwin\home\PETSC-~1.3-P\src\ksp\ksp\INTERF~1\itfunc.c
> [0]PETSC ERROR: KSPMonitorTrueResidualNorm() line 258 in 
> src/ksp/ksp/interface/C:\cygwin\home\PETSC-~1.3-P\src\ksp\ksp\INTERF~1\iterativ.c
> [0]PETSC ERROR: GMREScycle() line 155 in 
> src/ksp/ksp/impls/gmres/C:\cygwin\home\PETSC-~1.3-P\src\ksp\ksp\impls\gmres\gmres.c
> [0]PETSC ERROR: KSPSolve_GMRES() line 241 in 
> src/ksp/ksp/impls/gmres/C:\cygwin\home\PETSC-~1.3-P\src\ksp\ksp\impls\gmres\gmres.c
> [0]PETSC ERROR: KSPSolve() line 379 in 
> src/ksp/ksp/interface/C:\cygwin\home\PETSC-~1.3-P\src\ksp\ksp\INTERF~1\itfunc.c
> [0]PETSC ERROR: SNES_KSPSolve() line 2578 in 
> src/snes/interface/C:\cygwin\home\PETSC-~1.3-P\src\snes\INTERF~1\snes.c
> [0]PETSC ERROR: SNESSolve_LS() line 184 in 
> src/snes/impls/ls/C:\cygwin\home\PETSC-~1.3-P\src\snes\impls\ls\ls.c
> [0]PETSC ERROR: SNESSolve() line 1871 in 
> src/snes/interface/C:\cygwin\home\PETSC-~1.3-P\src\snes\INTERF~1\snes.c
> [0]PETSC ERROR: User provided function() line 574 in 
> unknowndirectory/c:\cygwin\home\rabbit\pgfem - csrbf\domain.cpp
> [0]PETSC ERROR: User provided function() line 40 in 
> unknowndirectory/c:\cygwin\home\rabbit\pgfem - csrbf\main.cpp
>
>
> Rong
>
> ----- Original Message ----- 
> From: "Barry Smith" <bsmith at mcs.anl.gov>
> To: "PETSc users list" <petsc-users at mcs.anl.gov>
> Sent: Wednesday, May 11, 2011 11:05 AM
> Subject: Re: [petsc-users] nonzero prescribed boundary condition
>
>
>
>   Ok, the linear solver looks like it is working ok. The likely problem is
> that Jacobian does not match the function evaluation.
>
>   Run the same thing but with the additional option -snes_mf_operator
>
>   Then run with -snes_type test (instead of -snes_mf_operator).
>
>   Barry
>
> On May 10, 2011, at 8:14 PM, Tian(ICT) wrote:
>
>> Dear Barry,  here is the output using -pc_type
>> lu -ksp_monitor_true_residual -snes_monitor -ksp_monitor
>> the attached is the same and for clear reference. Thanks again for helps.
>>
>> atol=1e-050, rtol=1e-008, stol=1e-008, maxit=50, maxf=10000
>> 0 SNES Function norm 7.071067811865e-002
>> 0 KSP Residual norm 9.965778978387e-002
>> 0 KSP preconditioned resid norm 9.965778978387e-002 true resid norm
>> 7.071067811865e-002 ||Ae||/||Ax|| 1.000000000000e+000
>> 1 KSP Residual norm 6.823187455811e-017
>> 1 KSP preconditioned resid norm 6.823187455811e-017 true resid norm
>> 8.847298885656e-011 ||Ae||/||Ax|| 1.251197007446e-009
>> 1 SNES Function norm 6.401926523423e-002
>> 0 KSP Residual norm 8.969200212486e-002
>> 0 KSP preconditioned resid norm 8.969200212486e-002 true resid norm
>> 6.401926523423e-002 ||Ae||/||Ax|| 1.000000000000e+000
>> 1 KSP Residual norm 1.106757475780e-016
>> 1 KSP preconditioned resid norm 1.106757475780e-016 true resid norm
>> 6.211830067439e-011 ||Ae||/||Ax|| 9.703063671087e-010
>> 2 SNES Function norm 5.849992149767e-002
>> 0 KSP Residual norm 8.072279488157e-002
>> 0 KSP preconditioned resid norm 8.072279488157e-002 true resid norm
>> 5.849992149767e-002 ||Ae||/||Ax|| 1.000000000000e+000
>> 1 KSP Residual norm 1.268750073799e-017
>> 1 KSP preconditioned resid norm 1.268750073799e-017 true resid norm
>> 3.802431036387e-011 ||Ae||/||Ax|| 6.499890835816e-010
>> 3 SNES Function norm 5.376618503592e-002
>> 0 KSP Residual norm 7.265050969883e-002
>> 0 KSP preconditioned resid norm 7.265050969883e-002 true resid norm
>> 5.376618503592e-002 ||Ae||/||Ax|| 1.000000000000e+000
>> 1 KSP Residual norm 2.677655733356e-017
>> 1 KSP preconditioned resid norm 2.677655733356e-017 true resid norm
>> 8.120397788686e-011 ||Ae||/||Ax|| 1.510316899602e-009
>> 4 SNES Function norm 4.956894354459e-002
>> 0 KSP Residual norm 6.538545411661e-002
>> 0 KSP preconditioned resid norm 6.538545411661e-002 true resid norm
>> 4.956894354459e-002 ||Ae||/||Ax|| 1.000000000000e+000
>> 1 KSP Residual norm 9.557004153175e-017
>> 1 KSP preconditioned resid norm 9.557004153175e-017 true resid norm
>> 2.944250802029e-011 ||Ae||/||Ax|| 5.939708598754e-010
>> 5 SNES Function norm 4.575418613137e-002
>> 0 KSP Residual norm 5.884690496914e-002
>> 0 KSP preconditioned resid norm 5.884690496914e-002 true resid norm
>> 4.575418613137e-002 ||Ae||/||Ax|| 1.000000000000e+000
>> 1 KSP Residual norm 5.470969262115e-017
>> 1 KSP preconditioned resid norm 5.470969262115e-017 true resid norm
>> 3.659003166095e-011 ||Ae||/||Ax|| 7.997089393284e-010
>> 6 SNES Function norm 4.223022245585e-002
>> 0 KSP Residual norm 5.296221144636e-002
>> 0 KSP preconditioned resid norm 5.296221144636e-002 true resid norm
>> 4.223022245585e-002 ||Ae||/||Ax|| 1.000000000000e+000
>> 1 KSP Residual norm 8.255198782390e-017
>> 1 KSP preconditioned resid norm 8.255198782390e-017 true resid norm
>> 1.955545658933e-011 ||Ae||/||Ax|| 4.630678090739e-010
>> 7 SNES Function norm 3.894430065910e-002
>> 0 KSP Residual norm 4.766598785088e-002
>> 0 KSP preconditioned resid norm 4.766598785088e-002 true resid norm
>> 3.894430065910e-002 ||Ae||/||Ax|| 1.000000000000e+000
>> 1 KSP Residual norm 3.322615478395e-017
>> 1 KSP preconditioned resid norm 3.322615478395e-017 true resid norm
>> 3.485328148673e-011 ||Ae||/||Ax|| 8.949520442496e-010
>> 8 SNES Function norm 3.586683371135e-002
>> 0 KSP Residual norm 4.289938708067e-002
>> 0 KSP preconditioned resid norm 4.289938708067e-002 true resid norm
>> 3.586683371135e-002 ||Ae||/||Ax|| 1.000000000000e+000
>> 1 KSP Residual norm 6.181358328498e-017
>> 1 KSP preconditioned resid norm 6.181358328498e-017 true resid norm
>> 3.246902818086e-011 ||Ae||/||Ax|| 9.052660862724e-010
>> 9 SNES Function norm 3.298130202025e-002
>> 0 KSP Residual norm 3.860944676473e-002
>> 0 KSP preconditioned resid norm 3.860944676473e-002 true resid norm
>> 3.298130202025e-002 ||Ae||/||Ax|| 1.000000000000e+000
>> 1 KSP Residual norm 4.635174776374e-017
>> 1 KSP preconditioned resid norm 4.635174776374e-017 true resid norm
>> 1.497516842272e-011 ||Ae||/||Ax|| 4.540502498513e-010
>> 10 SNES Function norm 3.027806208930e-002
>> 0 KSP Residual norm 3.474850078591e-002
>> 0 KSP preconditioned resid norm 3.474850078591e-002 true resid norm
>> 3.027806208930e-002 ||Ae||/||Ax|| 1.000000000000e+000
>> 1 KSP Residual norm 2.389914053685e-017
>> 1 KSP preconditioned resid norm 2.389914053685e-017 true resid norm
>> 6.007440888596e-011 ||Ae||/||Ax|| 1.984090286517e-009
>> 11 SNES Function norm 2.749422924729e-002
>> 0 KSP Residual norm 3.081350823297e-002
>> 0 KSP preconditioned resid norm 3.081350823297e-002 true resid norm
>> 2.749422924729e-002 ||Ae||/||Ax|| 1.000000000000e+000
>> 1 KSP Residual norm 2.640567497647e-017
>> 1 KSP preconditioned resid norm 2.640567497647e-017 true resid norm
>> 1.281638295853e-011 ||Ae||/||Ax|| 4.661481085089e-010
>> 12 SNES Function norm 2.437488247885e-002
>> 0 KSP Residual norm 2.633007441879e-002
>> 0 KSP preconditioned resid norm 2.633007441879e-002 true resid norm
>> 2.437488247885e-002 ||Ae||/||Ax|| 1.000000000000e+000
>> 1 KSP Residual norm 2.772331460094e-017
>> 1 KSP preconditioned resid norm 2.772331460094e-017 true resid norm
>> 1.918212496143e-011 ||Ae||/||Ax|| 7.869627670236e-010
>> 13 SNES Function norm 2.079664278637e-002
>> 0 KSP Residual norm 2.104738289397e-002
>> 0 KSP preconditioned resid norm 2.104738289397e-002 true resid norm
>> 2.079664278637e-002 ||Ae||/||Ax|| 1.000000000000e+000
>> 1 KSP Residual norm 1.650632708670e-017
>> 1 KSP preconditioned resid norm 1.650632708670e-017 true resid norm
>> 2.316371967362e-011 ||Ae||/||Ax|| 1.113820144509e-009
>> 14 SNES Function norm 1.657344626858e-002
>> 0 KSP Residual norm 1.454141853505e-002
>> 0 KSP preconditioned resid norm 1.454141853505e-002 true resid norm
>> 1.657344626858e-002 ||Ae||/||Ax|| 1.000000000000e+000
>> 1 KSP Residual norm 1.129401160070e-017
>> 1 KSP preconditioned resid norm 1.129401160070e-017 true resid norm
>> 7.885499327559e-012 ||Ae||/||Ax|| 4.757911661686e-010
>> 15 SNES Function norm 1.484243752612e-002
>> 0 KSP Residual norm 5.241948491751e-009
>> 0 KSP preconditioned resid norm 5.241948491751e-009 true resid norm
>> 1.484243752612e-002 ||Ae||/||Ax|| 1.000000000000e+000
>> 1 KSP Residual norm 2.729506849025e-024
>> 1 KSP preconditioned resid norm 2.729506849025e-024 true resid norm
>> 6.386677851085e-018 ||Ae||/||Ax|| 4.302984492839e-016
>> 16 SNES Function norm 2.828002157497e-008
>> 0 KSP Residual norm 6.042518362322e-015
>> 0 KSP preconditioned resid norm 6.042518362322e-015 true resid norm
>> 2.828002157497e-008 ||Ae||/||Ax|| 1.000000000000e+000
>> 1 KSP Residual norm 6.272441346127e-030
>> 1 KSP preconditioned resid norm 6.272441346127e-030 true resid norm
>> 1.112857698032e-023 ||Ae||/||Ax|| 3.935137372797e-016
>> 17 SNES Function norm 2.960967020289e-008
>> STEP 0 (Newton iterations: 17)
>>
>> diverged reason: -6
>>
>>
>> ----- Original Message ----- From: "Barry Smith" <bsmith at mcs.anl.gov>
>> To: "PETSc users list" <petsc-users at mcs.anl.gov>
>> Sent: Wednesday, May 11, 2011 2:54 AM
>> Subject: Re: [petsc-users] nonzero prescribed boundary condition
>>
>>
>>
>>  Use -pc_type lu -ksp_monitor_true_residual -snes_monitor -ksp_monitor 
>> and
>> send the outputs
>>
>>
>>  Barry
>>
>> On May 9, 2011, at 10:43 PM, Tian(ICT) wrote:
>>
>>> by the way, the increment size is like that
>>> for a 100 lengh model, the increment is set to 0.05,
>>> the engineering strain is around 5%% per load step.
>>> This is already too small increment size for a large deformation
>>> analysis.
>>> a 0.5 increment size leads to both linear search and trust region 
>>> failed.
>>> linear search failed for 0.05 while trust region converges with 17 
>>> Newton
>>> iterations each load step.
>>> Rong
>>>
>>> ----- Original Message ----- From: "Tian(ICT)" <rongtian at ncic.ac.cn>
>>> To: "PETSc users list" <petsc-users at mcs.anl.gov>
>>> Sent: Tuesday, May 10, 2011 11:37 AM
>>> Subject: Re: [petsc-users] nonzero prescribed boundary condition
>>>
>>>
>>>> First, thanks again, the issue was gone.
>>>>
>>>> I just followed up with some test results.
>>>> I have tested SNES using one finite element for a geometric large
>>>> deformation problem.
>>>> Those are just the very early test results so they may be not telling
>>>> what happened exactly.
>>>> For the displacement controlled load, I found that convergence is much
>>>> slower than that of force loading.
>>>> Even worse, linear search is so sensitive to the displacement increment
>>>> and diverged no matter what the increment size was used (too small
>>>> incremnt also led to diverged soloution (-6 reason), trust region works
>>>> well in the sense of not sensitive to the displacement increment, but
>>>> during each load step, it requires around ten to several tens of Newton
>>>> interations whereas for the force loading case and the almost same
>>>> amount of deformation, this is normally 3. This is against my
>>>> expectation. Any hint?
>>>>
>>>> Rong
>>>>
>>>> ----- Original Message ----- From: "Barry Smith" <bsmith at mcs.anl.gov>
>>>> To: "PETSc users list" <petsc-users at mcs.anl.gov>
>>>> Sent: Tuesday, May 10, 2011 10:22 AM
>>>> Subject: Re: [petsc-users] nonzero prescribed boundary condition
>>>>
>>>>
>>>>
>>>> On May 9, 2011, at 9:15 PM, Tian(ICT) wrote:
>>>>
>>>>> Dear Barry, Thanks a lot for quick answering.
>>>>> I checked the development documents and found the new version of
>>>>> MatZeroRows() does support the nonzero prescribed boundary conditions.
>>>>>
>>>>> I followed up with more details.
>>>>> I am using Petasc 2.3.3. to solve a nonlinear problem, e.g. using SNES
>>>>> solvers.
>>>>> I used a displacement-controlled load (as this type of loading works
>>>>> well for all cases).
>>>>> This is the reason the nonzero prescribed boundary came up.
>>>>>
>>>>> In FormJacobian, I modified Jacobian and residual to satisfy the
>>>>> nonzero prescribed boundary.
>>>>> In FormFunction, I modified the solution to the known solution(this
>>>>> should not be necessary as the modified Jacobian and rhs should give
>>>>> the prescribed solution also)
>>>>
>>>> You should not do it this way. See below.
>>>>>
>>>>> Now I found another issue, no matter if I prescried the solution or 
>>>>> not
>>>>> in FormFunction,
>>>>> SNES solver always call FormFunction and never call FormJacobian.
>>>>
>>>> The only reason it would not call FormJacobian is if decided that the
>>>> residual norm was small enough before any Newton steps; for example if
>>>> the FormFunction() computed exactly the zero function initially. When
>>>> you run with -snes_monitor -ksp_monitor what does it print for residual
>>>> norms.
>>>>
>>>>> Of course the solver finally diverged or converged to a zero solution.
>>>>>
>>>>> So my quick follow up question is How a displacement-controled load is
>>>>> done corrently in Petsc 2.3.3?
>>>>
>>>> To do it in 2.3.3 simply have for those components of F() the formula
>>>> F_i = x_i - givenvalue_i and in your Jacobian just use MatZeroRows() 
>>>> for
>>>> those rows
>>>>
>>>> We strongly urge you to upgrade to the latest PETSc before doing
>>>> anything further.
>>>>
>>>>
>>>>  Barry
>>>>
>>>>>
>>>>> Rong
>>>>>
>>>>> ----- Original Message ----- From: "Barry Smith" <bsmith at mcs.anl.gov>
>>>>> To: "PETSc users list" <petsc-users at mcs.anl.gov>
>>>>> Sent: Tuesday, May 10, 2011 9:31 AM
>>>>> Subject: Re: [petsc-users] nonzero prescribed boundary condition
>>>>>
>>>>>
>>>>>
>>>>> In petsc-dev
>>>>> http://www.mcs.anl.gov/petsc/petsc-as/developers/index.html we have
>>>>> modified the calling sequence for MatZeroRows() so that it can
>>>>> automatically adjust the appropriate right hand side values for the
>>>>> zeroed rows to support zero or non-zero prescribed boundary conditions
>>>>> easily.
>>>>>
>>>>> Barry
>>>>>
>>>>> On May 9, 2011, at 8:18 PM, Tian(ICT) wrote:
>>>>>
>>>>>> Dear all,
>>>>>>
>>>>>> I got this question long ago and searched the prior posting but did
>>>>>> not find the solution.
>>>>>> The question is about nonzero prescribed boundary condition.
>>>>>> My understanding is that MatZeroRows() works only for zero prescribed
>>>>>> value, not non-zero value.
>>>>>> For the non-zero values, we have to remove the rows associated with
>>>>>> the boundary, but this
>>>>>> will lead to a zero dignal and accordingly the rows in r.h.s should
>>>>>> also be removed.
>>>>>> My question is that does MatZeroRows() also works for nonzero
>>>>>> prescribed boundary and if so how to do it simply?
>>>>>>
>>>>>> Rong
>>>>>
>>>>>
>>>>
>>>>
>>>
>>
>> <aa>
>
>
> 