Abstract
We present an efficient and robust method which performs well for both strain limiting and treatment of simultaneous collisions. Our method formulates strain constraints and collision constraints as a serial of linear matrix inequalities (LMIs) and linear polynomial inequalities (LPIs), and solves an optimization problem with standard convex semidefinite programming solvers. When performing strain limiting, our method acts on strain tensors to constrain the singular values of the deformation gradient matrix in a specified interval. Our method can be applied to both triangular surface meshes and tetrahedral volume meshes. Compared with prior strain limiting methods, our method converges much faster and guarantees triangle flipping does not occur when applied to a triangular mesh. When performing treatment of simultaneous collisions, our method eliminates all detected collisions during each iteration, leading to higher efficiency and faster convergence than prior collision treatment methods.
Highlights
To prohibit excessive extensibility in simulation, most dynamic models use projection to enforce a hard limit on large strains, i.e., strain limiting
To solve our problem, we adopt the method proposed by Kovalsky et al [20], which can control the singular values of a square matrix to lie within a positive interval [γ, Γ ] by use of semidefinite programming (SDP)
Strain and collision constraints can be reformulated as linear matrix inequalities (LMIs) and linear polynomial inequalities (LPIs), allowing us to take advantage of standard convex SDP solvers to solve strain limiting and collision problems in physically based simulation
Summary
Strain limiting for a single triangle can be defined via the following projection optimization problem with LMI constraints: Our method Narain et al.’s method. After transforming the triangle into a tetrahedron, F is a square matrix. Because F is just a linear transformation of q, we can apply the method to our strain limiting problem. Having shown how strain limiting for a single triangle or tetrahedron is done, it is easy to extend Eq (18) to the case for multiple triangles and tetrahedra
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