Re: Lagrange mutlipliers and solution methods
From: Peter Spellucci (spellucci_at_fb04373.mathematik.tu-darmstadt.de)
Date: 03/23/05
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Date: Wed, 23 Mar 2005 15:58:29 +0000 (UTC)
In article <d1s2lf$qe5$1@arcturus.ciril.fr>,
Bob <none@none.com> writes:
>Thanks a lot for this detailed information and valuable pointer.
>Actually I am investigating a tied contact in elastostatics algorithm
>(e.g. relative movement of fault blocks) and my understanding is that
>there is no inequalities. Furthermore, I do have multi-freedoms
>constraints that require usage of Lagrange mutipliers (equalities too).
>
>I read (partially, I must admit) the web site and the LP, it seams still
>to fit my problem. I however having a bad time to go from my A x = b
f(x) = (1/2)x^T A x - b^Tx => grad f(x) = Ax-b
Ax-b=0 woulb be the unconstrained minimum, A is the stiffness matrix and b the
load.
if you incorporate equality constraints (nodes being in contact) then you get an
augmented system
[ A H ] [x ] = [b ]
[ H^T O ] [my] [-h0]
where the equality constraints (a set of linear equalities) are expressed as
H^Tx+h0 = 0
my is the vector of Lagrange multipliers. if the condition number is not too bad
you could solve this by minres or symmlq iteratively. there are also known
preconditioners for indefinite symmetric systems.
but in general, should there be constraints of the type
boundary of bdoy one must not cut the boundary of body 2 must not cut ...
resultiong in general linear inequalites with respect to the nodes if you
have a (linear) triangularization/tetrahization or even quadratic cnstraints if
you have isoparametric quadratic elements?
>linear system to the objective function (A being stifness matrix
>augmented with constraints in Column Compressed Storage).
>
>Anyway,
>
>Thanks
>
>
>Peter Spellucci wrote:
>
>> Bob <none@none.com> writes:
>> >Hi all, I am trying to be more familiar with computational contact
>> >mechanics, it makes intensive use of lagrange multipliers (and derived
>> >forms). We may loss posite definiteness of the system (CG is out).
>> >
>> >- From a theoritical point of view*, what class of solver may I use?
>> >Pointers (www/books) would be nice.
>> >
>> >- From a computational point of view, is there techniques to by-pass the
>> > problem? Is there any C/C++/Fortran library that will provide
>> >necessary solvers?
>> >
>> >Thanks.
>> >
>> >Bob
>> >
>> >* Feel free to go back to basics, I am a geologist with few mathematical
>> >background
>> >
>>
>> contact problems become after discretization by the method of finite elements
>> (the usual discretization method here) large scale nonlinear optimization problems
>> with inequality constraints (and maybe additional equality constraints).
>> but usually they are not dealt with as optimization problems "from scratch"
>> since that combination of large scale with a lot of general and possibly even
>> nonlinear inequality constraints make them extremely expensive.
>> In your case, there obviously convexity (=definiteness) is lost, the things become
>> even worse. the usual way to deal with these problems is to solve them by a
>> continuation method applied to the necessary optimality conditions, beginning
>> with zero contact (zero external force) and then applying more and more external
>> force until the actual configuration is reached.
>> if you want to go from scratch in the nonconvex case you will also face the
>> problem of local but nonglobal minimizer which may be physically senseless.
>> you find software for large scale optimization at
>> http://plato.la.asu.edu/topics/problems/nlores.html
>> and continuation methods for nonlinear systems (in your case the
>> Kuhn-Tucker-conditions) in
>> http://plato.la.asu.edu/topics/problems/zero.html
>> you may also consult the subpage
>> http://plato.la.asu.edu/topics/problems/mcp.html
>> since the Kuhn-Tucker-conditions represent a special case of these.
>> hth
>> peter
>>
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