The method of force density was developed in response to the need for computational modelling of structures for the Munich Olympic complex. The method relies on the assumption that the ratio of tension force to length of each cable can be constant, transforming a system of non-linear equations to a set of linear equations which can be solved directly. In this short note I will give an introduction, description and some results of using the FDM for use as a parameterization technique

Southern, Richard

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Bournemouth University Research Online

Technical Report TR-NCCA-2011-02THE FORCE DENSITY METHOD: A BRIEFINTRODUCTIONRichard SouthernThe National Centre for Computer AnimationBournemouth Media SchoolBournemouth UniversityTalbot Campus,Poole, Dorset BH12 5BBUnited Kingdom2011Technical Report TR-NCCA-2011-02ISBN 1-85899-123-4Title The Force Density Method: A Brief IntroductionAuthor(s) Richard SouthernKeywords Parameterization, Force Density MethodAbstractThe method of force density was developed in response to the need for computational mod-elling of structures for the Munich Olympic complex[Lewis, 2003]. The method relies on theassumption that the ratio of tension force to length of each cable can be constant, transform-ing a system of non-linear equations to a set of linear equations which can be solved directly.In this short note I will give an introduction, description and some results of using the FDMfor use as a parameterization technique.Report date 18 May 2011Web site to download from http://eprints.bournemouth.ac.uk/The authors’ e-mail addressesrsouthern@bmouth.ac.ukSupplementary notesUnpublishedThe National Centre for Computer AnimationBournemouth Media SchoolBournemouth UniversityTalbot Campus,Poole, Dorset BH12 5BBUnited KingdomThe Force Density Method without the agonizing painRichard SouthernWednesday 18th May, 20111 IntroductionThe method of force density was developed in re-sponse to the need for computational modelling ofstructures for the Munich Olympic complex[Lewis,2003]. The method relies on the assumption thatthe ratio of tension force to length of each cable canbe constant, transforming a system of non-linearequations to a set of linear equations which can besolved directly.The Force Density Method (FDM), first intro-duced by [Schek, 1974], is commonly used in engi-neering to find the equilibrium shape of a struc-ture consisting of a network of cables with dif-ferent elasticity properties when stress is applied.While shape analysis of tensile structures is a geo-metrically non–linear problem, the FDM linearisesthe form–fitting equations analytically by using theforce density ratio for each cable element, q = F/L,where F and L are the force and length of a cableelement respectively.2 The Force Density MethodGiven the positions of nodes (vertices) which con-nect the cables (edges) of our network V , the topol-ogy of this network is encoded in the branch–nodematrix C (see Figure 1. Given a load vector R andthe diagonal matrix of force densities Q, the equi-librium location can be deduced by solving for Xin(CTQC)X = R. (1)The FDM has certain properties which makes itof interest in computer graphics:• it represents a minimum energy surface, andtherefore• it approximates a C2 continuous surface,• it depends only on the force density of theedges and the topology of the network, and• the system is sparse, symmetric and positivedefinite, quickly solved using the conjugategradient method.Cf C1 2 3 4 5 6 7 8a 1 . . -1 . . . .b . 1 -1 . . . . .c . . 1 -1 . . . .d 1 . . -1 . . . .e 1 . . . -1 . . .f . 1 . . . -1 . .g . . 1 . . . -1 .h . . . 1 . . . -1i . . . . 1 -1 . .j . . . . . 1 -1 .k . . . . . . 1 -1l . . . . 1 . . -1Figure 1: Deriving the branch–node matrix. Eachrow corresponds to an edge, while columns corre-spond to nodes. Note that the order in which thenodes are specified on the edge will not affect thesystem.3 EmbeddingIn order to use the FDM for 2D embedding, wemust be able to constrain nodes on the bound-ary. For this, the matrix C is separated into twosub–matrices: Cf contains constrained nodes withcorresponding position Xf , while C contains thosethat are free to move. The problem in Equation 1is reformulated as(CTQC)X = R− (CTQCf )Xf . (2)1Note that for the purposes of embedding, R is typ-ically set to zero.The FDM is fold–over free. This is explainedwith reference to Figure 2. As the natural rest in-ternal force load of each node is 0, any foldover willresult in external forces applied to the nodes whichhave folded over. As a result it will not be in astate of equilibrium, and this configuration cannotoccur.Figure 2: Foldover with FDM cannot occur, as itwill not be in a state of equilibrium.4 Stability under motionThe FDM cannot guarantee any of the commonlyadvocated properties of embeddings in computergraphics, such as isometry (length preserving) orconformality (angle preserving). In our applica-tion, it is desirable that the path of a vertex ora group of vertices moving across the surface ofthe shape is mirrored in the embedding. We evalu-ate this phenomenon by measuring the distortion ofthese displacement vectors in the embedded space.We call this property stability under motion.In Figure 3, we compare two popular fixedboundary conformal techniques, Harmonic map-pings [Eck et al., 1995] and Mean Value Coordi-nates [Floater, 2003] with the FDM with unit edgeforces. The stability under motion of these em-bedding methods is highly non–linear, and so weevaluate each embedding technique experimentallyas follows:1. Compute the embedding U0 = embed(M0).2. Rotate a set of points on the sphere (in thiscase, one triangle) in a straight path aroundthe surface of the sphere.3. For each state of the rotation Mi computeUi = embed(Mi).Harmonic Mean Value FDM[Eck et al., 1995] [Floater, 2003] [Schek, 1974]Figure 3: A comparison of the stability under mo-tion of three embedding techniques.4. Compute the n displacement vectors in theembedded space. So for each u0,i ∈ U0 andud,i ∈ Ud define di as the vector between u0,iand ud,i, andD = [d1, . . . ,dn]T.5. We use the angle of the normalcone of these displacement vectorsα = maxi,j (acos(di · dj)) as the distor-tion error. For this experiment, we evaluatethe distortion of only the points moving onthe surface.Figure 3 demonstrates that under all rotationsthe FDM embedding is stable, even when faces ofMd overlap. In addition, there is considerably lessdisplacement of surrounding nodes after rotations.5 ConclusionI have presented the force density method as a tech-nique to perform mesh embedding using this tech-nique. It is computationally efficient, as it onlyinvolves the solution to a sparse linear system, eas-ily solved with the Conjugate Gradient Method orusing Cholesky Factorization. In Figure 3 I demon-strate that embedding with FDM is very stable,2preventing foldover and discontinuities when pa-rameterizing an unstable triangulation. This is par-ticularly useful when, for example, you need to flat-ten geometry in a stable fashion (for example, seeZhang et al. [2007]).ReferencesM. Eck, T. D. DeRose, T. Duchamp, H. Hoppe,M. Lounsbery, and W. Stuetzle. Multiresolutionanalysis of arbitrary meshes. Proceedings of SIG-GRAPH, pages 173–182, 1995.Michael S. Floater. Mean value coordinates.Computer Aided Geometric Design, 20(1):19–27,2003.Wanda J. Lewis. Tension Structures: Form andBehaviour. Thomas Telford, illustrated edition,2003. ISBN 0727732366, 9780727732361.J. H. Schek. The force density method for form find-ing and computation of general networks. Com-puter Methods in Applied Mechanics and Engi-neering, (3):115–134, 1974.J. J. Zhang, X. Yang, and Y. Zhao. Bar–net drivenskinning for character animation. Computer Ani-mation and Virtual Worlds, 18(10):4–5, Septem-ber 2007.3

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