# the augmented global stiffness matrix is not positive semi-definite using Lagrange Multipliers method within FEM

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: 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?

• Lagrange multipliers lead to saddle point/indefinite systems; this is how it is supposed to be. Nov 14 '17 at 12:01
• 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 Nov 14 '17 at 13:37
• 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? Nov 14 '17 at 13:41
• I don't understand the formula in the question. What are the various symbols supposed to mean? Nov 15 '17 at 4:29
• 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. Nov 15 '17 at 6:03