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The augmented global stiffness matrix is not positive semi-definite when using Lagrange Multipliers method to enforce boundary constraints on a simple square domain of integral form:enter image description here I am considering linear elastic material constitutive. I checked the condition number which is order 10e3 indicating the augmented matrix is not much ill-conditioned. However, there are some negative eigenvalues for the matrix. What are possible reasons for the lack of semi-definite positiveness? It should be always positive semi-definite for the (linearized) global stiffness matrix whatever Lagrange multipliers method is employed. Is my Lagrange multipliers discretization/approximation space is not well set up or I lost something very important? Anyone can share some experience?

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    $\begingroup$ Lagrange multipliers lead to saddle point/indefinite systems; this is how it is supposed to be. $\endgroup$
    – Nick Alger
    Nov 14 '17 at 12:01
  • $\begingroup$ Yes. I was just impressed and misled by an online statement that the matrix from Lagrange multipliers formulation should be at least positive semidefinite. However, it is not the case in fact $\endgroup$ Nov 14 '17 at 13:37
  • $\begingroup$ Using direct solvers, I can obtain a solution. I was using NR FE solver for solution, the residual doesnot converge to some target! I wanna know whether I can use the scheme K*delta_u=residual for solution??? Why the force error is divergent so badly? $\endgroup$ Nov 14 '17 at 13:41
  • $\begingroup$ I don't understand the formula in the question. What are the various symbols supposed to mean? $\endgroup$ Nov 15 '17 at 4:29
  • $\begingroup$ Hi. Wolfgang. This is used to apply a macro strain/deformation to square RVE in a weak (integral) sense. x is the boundary point coordinates. Epsilon is the macro strain to enforce. u is the displacement. m is the RVE boundary index. Symbol j denotes the point of subsegments, because I subdivide the boundary into some zones and apply a Lagrange multiplier for each zone, then do the integration. $\endgroup$ Nov 15 '17 at 6:03

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