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FEBio is designed to solve problems that can undergo large deformations. However, when the deformations become too large it can happen that the finite element mesh cannot be distorted any further without inverting elements. In that case, FEBio will not be able to continue the solution and will most likely terminate with a negative jacobian error message.
One possible solution to this problem is to remesh the model at the point at which the deformation become too large, but currently FEBio does not have any remeshing capabilities. The only remedy at this point is to plan ahead and design your mesh in such a way that the elements will not invert in the range of deformation that you are interested in (e.g. place a butterfly mesh in sharp corners or make the mesh finer in areas of larger deformation).