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Kawamura et al. applied a spring–mass-shock absorbermodel for dynamic simulation of organs in MIS surgery [16]. Eventhough most of the work found in the literature is to simulate thedeformation of soft-tissues, it has been also applied to the process-ing ofmedical images. In thework presented byMatuszewski et al.a spring–mass systemis used to performthe non-rigid registrationof medical images that represent both rigid and deformable struc-tures [20].Other application in which spring–mass systems have beenused is facial animation. Kahler et al. used a spring–mass witha layer model for dynamic simulation of the human face [17].The layers represent different tissues simulated such as: bone,muscle, and skin. The spring–mass system is used to unify thedifferent layers of the model. In general, Vassilev and Spanlangproposed a spring–mass system to simulate deformable solids inreal time [21]. In this work, a new type of spring called supportingspring is introduced to guaranty volume preservation.Another application in which significant advances have beenmade using spring models is clothing simulation. Ji et al. useda squared spring–mass mesh and introduced a new method todetect collisions in order to simulate the behavior of cloths whenthey get in touch with other solid objects or on itself [18]. Asimilar technique was proposed by Yeung et al. [19]. They usedspring–mass meshes to model the flattening of surfaces thatcover 3D objects. In addition, Bridson et al. used a model withspring–mass in which the folding of the fabric is taken intoaccount [5]. They define a folding energy between triangularelements to model sheets. A similar expression for the foldingenergy was developed in a work presented by Grinspun et al. [6].In their work, they propose a general form for thin objects such asfabrics, paper, or sheet metal.In many applications spring–mass models are used to simulatethe behavior of different real-world objects without the need todefine the problem using continuous mechanics as with FEM. Itcan be applied where FEM simulation would be computationallyunfeasible for real-time applications. In addition, its implementa-tion is rather simple as well as its computational efficiency. Takinginto consideration these advantages, the present work proposes amethod to calculate deformations based on the use of spring–masssystem. The model is used to calculate the deformation of flexibleparts such as shell-type objects when they are fixed to a mechani-cal jig in order to be compared to a CAD model.It is relevant to mention that one of the main difficulties ofusing spring–mass systems is to determine the model parameterssuch as weight of the punctual masses and the spring stiffnessconstants. In most cases an empirical determination of theseparameters is performed. For example, Duysak et al. proposed amethod based on neural networks to determine the spring stiffnessand damping constants in a facial tissue simulation [22]. Similarly,Lloyd et al. introduce a method to calculate the spring stiffnessconstants by using isotropic elasticmodels as a Ref. [23]. Analyticalexpressions that allow the calculation of the stiffness constantsare derived from equations describing the material behavior usingFEM. Finally, San-Vicente et al. presented a method to determinethe parameters of volumetric spring–mass systems made of cubicelements; those parameters are used to model biological tissuesin interactive applications [24]. The method is based on theadjustment of analytical expressions resulting from the behaviorof testing models.The present work takes into consideration two parameters inorder to define the energy function of the system. The first one isthe material thickness, which is represented as a parameter thataffects the spring system uniformly.
The second parameter is theone defined locally according to the geometry of the polygonalmesh. In order to introduce such parameters, an experimentalanalysis was performed to simulate the behavior of a simplesystem by using FEM.3. A model for articulated springsThis section describes the physical–mathematical model pro-posed to calculate the deformation of virtual models for part in-spection. It assumes that the geometry of the parts is representedby a virtual model made of polygonal elements. The deformationmodeling involves, initially, a discrete geometrical representationof the object, and then, the introduction of constraints and thephysical laws that governs deformation.As for real elastic deformation, virtual objects must storepotential energy in its own structure as a result of deformation.We Fig. 1. Articulation of elastic plates.assume that the deformation energy is stored in a spring systemassociated to the polygonal mesh that represents the geometry ofthe object. Taking into account that the equilibrium position ofthe systemcorresponds tominimumenergy, the calculation of thepositions of the nodes in the deformed mesh is obtained using aniterative energy minimization algorithm.3.1. Geometry and constraintsA model is built to simulate the deformation of shell-type partswhose geometry can be represented approximately by a surface ina three-dimensional space. It is assumed that the inspected objectsare made of homogeneous and isotropic materials.For simplicity, the polygonal model that represents the objectis considered to be composed only by triangular elements. In orderto model the deformation of the polygonal surface, two types ofsprings are defined as well as a structure of articulations. In thismodel, it is considered that there is a spring that joins together thetwo nodes of the polygonal model. This set of springs is locatedon the same polygonal surface. We also consider that a spring isconnected to the opposite corners of each pair of adjacent facesthat share a common edge. This second set of springs is not part ofthe geometry of the polygonal model.Let {f1, f2} be a pair of adjacent triangular faces connected toeach other as an articulation, {n1, n2} the nodes of the commonedge that act as the articulation and {n3, n4} the opposite nodes(see Fig. 1). The first type of spring is defined by the connectionsbetween the pairs of nodes {n1, n2}, {n2, n3}, {n3, n1}, {n1, n4} and{n4, n2}. The second type of spring is represented by a connectionbetween nodes {n3, n4} which are the opposite nodes of thearticulation.Fig. 1 shows an articulation formed by two faces. Each of theedges of the triangular faces represents a spring, so each of thefaces is elastic and therefore can be deformed on the same plane.In other words, each triangular element behaves as a membrane.The external spring between the opposite nodes of the articulationallows to model bending behavior for a two-faces system.3.2. The energy functionIn the model, we assumed that for each spring in the model,the length change u is directly proportional to the longitudinalforce F applied to the spring. This relationship called Hooke’s Lawis expressed mathematically as the following:F = ku, (1)where is a proportionality constant k and is called the spring’sstiffness constant. When we have real springs, the value ofthis constant can be calculated by using different experimentalmethods [25,26]. Nevertheless, as in this case it is being dealt witha virtual springmodel, this work proposes a simulated experimentusing FEMfromwhich amathematical expression to determine theconstant value of the model stiffness is put forward. The energy of deformation E stored in a spring that followsHooke’s law (Eq. (1)) is expressed by:E = 12ku2. (2)Then, for a system made of n linear springs, the total energy ofthe system ET is expressed by the sum of the inpidual springs,ET = 12n i=1kiu2i. (3)Taking into account that the system is made by two typesof springs, one associated to the membrane behavior of thepolygonal mesh and the other one associated to the bendingbetween polygonal faces, the stiffness constants are noted in adifferentway: km formembrane springs and kf for bending springs.Separating the terms according to the stiffness constants, theenergy is expressed by:ET = 12ikm,iu2i+ 12jkf ,ju2j, (4)where i and j sub-indexes are for the membrane and the bendingsprings respectively.Let Em be the term for the energy associated to the membranebehavior and Ef be the termassociated to the bending behavior,wehave:ET = Em + Ef . (5)3.2.1. Membrane energySimilarly to thework of Lloyd et al. [23], in order to calculate thestiffness constant associated to the membrane behavior, a relationthat depends linearly on the model’s thickness h is proposed; thatis:km = αh, (6)where α is a constant of proportionality. Since it is assumed thatthe part has uniform thickness, the constant km is the same for allmembrane springs and, therefore, the energy of the membrane isexpressed by:Em = 12kmiu2i, (7)here, the sub-index i indicates the ith spring associated to thepolygonal mesh.3.2.2. Bending energyIn order to determine a mathematical expression to calculatethe stiffness constant associated to the bending behavior anexperiment was carried out using an FEM based simulator. Amodelmade of two triangular shell-type elements was used in theexperiment as illustrated in Fig. 2. The thickness h, the length of thecommon side b, and the average length of the opposite sides beforeapplying the deformation ˜ a,were used as parameters of themodel.A force F was applied at one end of the model while the other endwas fixed. Constraintswere imposed on themodel’s nodes in orderto give stability to the system.Initially with the model parameters fixed, applying differentvalues for F and calculating the displacement u which results ineach case to be a linear relationship between the two variables,as expressed in Eq. (1).