CS-184: Computer GraphicsLecture #21: Spring and Mass systemsProf. James O’BrienUniversity of California, BerkeleyV2006-F-21-1.02TodaySpring and Mass systemsDistance springsSpring dampersEdge springs3A Simple SpringIdeal zero-length springForce pulls points togetherStrength proportional to distance fa→b= ks(b − a)fb→a= −fa→b4A Simple SpringEnergy potentialfa→b= ks(b − a)fb→a= −fa→b-2-1012-2-101202468-012fa= −∇aE = −!∂E∂ax,∂E∂ay,∂E∂az"E = 1/2 ks(b − a) · (b − a)5A Simple SpringEnergy potential: kinetic vs elastic-6 -4 -2 2 4 6-1-0.50.51E = 1/2 ks(b − a) · (b − a)E = 1/2 m(˙b −˙a) · (˙b −˙a)6Non-Zero Length Springsfa→b= ksb − a||b − a||(||b − a|| − l)Rest lengthE = ks(||b − a|| − l)2-2-1012-2-10120123-0127Comments on SpringsSprings with zero rest length are linearSprings with non-zero rest length are nonlinerForce magnitude linear w/ discplacement (from rest length)Force direction is non-linearSingularity at ||b − a|| = 08Damping“Mass proportional” dampingBehaves like viscous drag on all motionConsider a pair of masses connected by a springHow to model rusty vs oiled springShould internal damping slow group motion of the pair?Can help stability... up to a pointf = −kd˙af˙a9Damping“Stiffness proportional” dampingBehaves viscous drag on change in spring lengthConsider a pair of masses connected by a springHow to model rusty vs oiled springShould internal damping slow group motion of the pair?fa= −kdb − a||b − a||2(b − a) · (˙b −˙a)10Spring ConstantsTwo ways to model a single springl∆l∆l/2l/2l/2∆l/211Spring ConstantsConstant gives inconsistent results with different discretizationsChange in length is not what we want to measureStrain: change in length as fraction of original lengthks! =∆ll0Nice and simple for 1D...12Structures from SpringsSheetsBlocksOthers13Structures from SpringsThey behave like what they are (obviously!)This structure will not resist shearingThis structure will not resist out-of-plane bending either...14Structures from SpringsThey behave like what they are (obviously!)This structure will resist shearingbut has anisotopic biasThis structure still will not resist out-of-plane bending15They behave like what they are (obviously!)Structures from SpringsThis structure will resist shearingLess biasInterference between spring setsThis structure still will not resist out-of-plane bending16They behave like what they are (obviously!)Structures from SpringsThis structure will resist shearingLess biasInterference between spring setsThis structure will resist out-of-plane bendingInterference between spring setsOdd behaviorHow do we set spring constants?17Edge SpringsBridson et al. / Simulation of Clothing4. An Accurate Model for BendingThe physics of cloth bending are poorly understood. The dy-namics of anisotropic fibers twined together and woven intoa sheet of fabric constantly interacting with massive defor-mations and friction is certainly more difficult to model witha two-dimensional continuum than for example steel. How-ever, several basic qualitative properties of such a model canbe identified that are essential for a plausible simulation, andwithout these a model is incorrect.In order to handle unstructured triangle meshes and getfiner, more robust control over bending than in vertex-centricmodels, we posit as our basic bending element two trian-gles sharing an edge. Our bending elements will be based onthe dihedral angle and its rate of change, as in Baraff andWitkin5and extended in Grinspun et al.24. We label the ele-ment as in figure 1, with vertex positions xiand velocities vi,i = 1,...,4, and angle θ between the normals n1and n2.The vector of the four velocities v = (v1,v2,v3,v4) and thevector of bending forces F = (F1,F2,F3,F4) live in a 12 di-mensional linear space. One can select a basis for this spaceidentifying twelve distinct “modes” of motion. For bendingit is natural to select for the first eleven modes the three rigidbody translations, the three (instantaneous) rigid body rota-tions, the two in-plane motions of vertex 1, the two in-planemotions of vertex 2, and the one in-line stretching of edge 3–4. None of these change the dihedral angle, and thus shouldnot participate in bending force calculations. This leaves thetwelfth mode, the bending mode, which is the unique modeorthogonal to the other eleven up to an arbitrary scaling fac-tor. This mode changes the dihedral angle but does not causeany in-plane deformation or rigid body motion. Let us callit u = (u1,u2,u3,u4). From the condition of orthogonalityto the in-plane motions of vertices 1 and 2, we find that u1is parallel to ˆn1and u2is parallel to ˆn2. From the conditionof orthogonality to the in-axis stretching of edge 3–4, wesee that u4− u3must be in the span of ˆn1and ˆn2. Orthog-onality to the rigid body translations implies that the sumu1+ u2+ u3+ u4is zero, and hence u3+ u4is also in thespan of ˆn1and ˆn2, thus u3and u4are each in this span. Fi-nally, after making u orthogonal to rigid rotations (which wecan conveniently choose to be about the axes ˆn1, ˆn2and ˆe)we end up with1234nen12^^^n1^n2^Figure 1: A bending element with dihedral angle π − θ.u1= |E|N1|N1|2u2= |E|N2|N2|2u3=(x1− x4) · E|E|N1|N1|2+(x2− x4) · E|E|N2|N2|2u4= −(x1− x3) · E|E|N1|N1|2−(x2− x3) · E|E|N2|N2|2up to an arbitrary scaling factor, where N1= (x1− x3) ×(x1− x4) and N2= (x2− x4)× (x2− x3) are the area weightednormals and E = x4− x3is the common edge. Thus u1andu2are inversely proportional to their distance from the com-mon edge, and u3and u4are a linear combination of u1andu2based on the barycentric coordinates of x1and x2with re-spect to the common edge. The bending elastic and dampingforces must be proportional to this mode. One immediate ob-servation is that orthogonality to rigid body modes impliesthese forces conserve linear and angular momentum. In fact,every bending model based on two triangles that does notuse exactly these force directions will violate either the fun-damental conservation laws or will influence in-plane (i.e.non-bending) deformations. While some may argue that inreality, in-plane and bending deformations are subtly cou-pled, the exact nature of this coupling varies between mate-rials and is not understood for even the simplest fabrics, thusit is wisest to avoid adding arbitrary and artificial coupling.For simplicity we choose the
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