Method and system of real-time graphical simulation of large rotational deformation and manipulation using modal warping

A system for real-time graphical simulation of large rotational deformation and manipulation of an object using modal warping comprising the modules for:

• a) mapping each material points of the object into a mesh of nodes of a predetermined number, n;
• b) decomposing a (infinitesimal) deformation of each material points into a strain and a rotation in terms of an infinitesimal strain tensor and infinitesimal rotation tensor;
• c) expressing a displacement field of each node as a linear combination of a basis of the 3n-dimensional space, wherein the basis of the 3n-dimensional space are the columns of a modal displacement matrix, a solution matrix (Φ) of the generalized eigenvalue problem (KΦ=MΦΛ) along with a diagonal matrix (Λ);
• d) taking only dominant m columns of the modal displacement matrix (Φ) by examining the eigenvalues in order to reduce the amount of computation;
• e) computing the motion components due to individual modes independently and combining the motion components by linear combination;
• f) expressing the rotational part (w(t)) of the motion of each node as a product of a modal rotation matrix (T) and the vector containing the modal amplitudes of the displacement field of each node, wherein the modal rotation matrix (T) is given by a product of a global matrix (W) and the modal displacement matrix(O), wherein the global matrix (W) is given such that the product of the global matrix (W) and the displacement field (u(t)) gives the composite vector (w(t)), the rotational part;
• g) precomputing the modal rotation matrix (Ψ);
• h) reflecting the rotations in the calculation of the displacement field (u(t)) by integrating the product of the rotation matrix (Ri(t)) and the differential displacement {dot over (u)}iL(t)dt over the time, wherein the rotation matrix represents the orientation of a local coordinate frame at time t, {i}(t) and the differential displacement of the i_th node is measured from {i}(t−dt) at time t−dt;
• i) reducing a linear elastodynamic equation in a local coordinate frames into a set of decoupled ordinary differential equations (ODEs);
• j) solving the decoupled ODEs numerically;
• k) evaluating analytically the integration of rotation to reflect into the displacement field by taking a quasi-static approach that ramps the modal amplitude vector (q(t)) to circumvent an hyteresis effect;
• and
• l) precomputing the time-invariant parts in the constraint equations in the position constraints and orientation constraints at the constraint initiation stage.