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Basically, the model has the information aboutthe coordinates of the nodes on the surface and their connections.For each triangular face, its corners represent the nodes of thepolygonal model and the edges represent the connections amongsuch nodes.Similar to what happens during the assembly of a real part,the deformation of a virtual model happens when fixation pointsbecome displaced. As opposed to a real part where the fixationpoints are the ones that are displaced to their assembly position, inthe proposed method the fixation points from the CAD model arethe ones displaced instead of the points in the real part. The mainadvantage of this approach is that in order to carry out the non-rigid alignment stage between both models, there is no need toacquire the whole surface of the inspected part using a 3D sensor.It is assumed that the CAD model represents the full model of thepart for a correct simulation of deformation.In this work, it is assumed that the two models to be comparedare: the data model and the CAD model and that they have beenpreviously aligned to each other by using a rigid transformation method. Such transformation can be calculated in a simple wayby taking as starting point the correspondences between fixationpoints of both models (Fig. 5). Even though, it is also usefulto carry out a fine alignment by using an iterative algorithmsuch as the Iterative Closest Point (ICP) algorithm [27]. Oncethe rigid transformation has been applied, the calculation of thedisplacements of the fixation points from the CAD model towardsthe data model is done. Those displacements are then applied tothe springmodel to propagate the deformation to the entiremodel(Fig. 6).4.1. Polygonal mesh reductionTaking into account that a dense polygonal mesh involves agreater computational burden than a simpler one, one can startthe inspection process by an optimization stage to reduce theinitial polygonalmesh. The only condition that prevails during thisprocess is that the fixation points on the CAD mesh need to bepreserved as nodes of the reduced polygonal model (Fig. 7). Thatis necessary as the fixation points are required to calculate thedeformation.In the literature, there are variousmethods proposed to performthis reduction process on triangular mesh in 3D. Many of themfollow the same principle which is to preserve the shape ofthe general model while eliminating nodes of the polygonal model [28–30]. Since the main interest in this work is to modeldeformation and not to simply reduce the model for visualization,a commercial software called PolyWorks/IMCompressR ⃝ [31] wasused to perform polygon reduction. The core of the program isan iterative algorithm that reduces the nodes of the triangularmodel while keeping the minimum distance between the reducedtriangulation and the original; in thatway, the shape of the originalCAD model is preserved while the number of triangular facesis reduced significantly. In order to ensure a low computationalburden when performing the optimization stage, the number ofnodes of the triangular meshes fluctuated between 20 and 30nodes during the tests. In most cases, the approximation valueneeds to be assigned by the users according to the need for speedand precision.4.2. Deformation of the reduced meshThe calculation of the final position of the nodes is performed onthe deformed mesh. Such calculation is carried out using an itera-tive optimization process inwhich the energy function ET (Eq. (10))is minimized.4.2.1. Initialization of the iterative processSince the iterative minimization process needs an initial valuefor the positions to be calculated, an interpolation technique basedon RBFs is applied to obtain it. The centers of the RBFs are thefixation points of the inspected part and the input values are thenodes of the reduced mesh. As explained in [4], the position of thecorresponding points of the deformedmesh F(pi) is obtained fromthe displacements of the control points qj and the position of the non-deformed mesh pi .
This calculation is carried out using thefollowing equation:F(pi) =jCjR(d) + pi, (14)where Cj is theweight associated to each center qj and R representsthe base function. In the present work, multi-quadrics are usedas basis functions: R(d) = (d2+ β2)1/2, where d(a, b) is theEuclidean distance between each pair of points (a, b) and β whichis a parameter calculated for every center qj as the minimumdistance to the other centers qk [32] as defined by:βj = minj̸=kdj(qk). (15)Fig. 8(a) shows an example of initialization on a bar model.In this example, control points are the extreme of the bar model.The orange curve represents the initialmeshwithout deformation.The interpolation with RBFs, using only the fixation points, isshown in green. The result of the optimization applying the springmodel is shown in blue. The obtained curve with the RBFs (green)represents an intermediate stage between the initial CAD (orange)and the final deformed model (blue). Fig. 8(b) shows again thedeformed spring–mass model but now it is compared to thedeformation obtained through an FEM simulation with beam-typeelements. In this figure, the green curve is very close to the blueone representing the deformed model using FEM. In this example,a reduction of the initial mesh is not applied, nevertheless, theinterpolation with RBFs is applied to generate the initial positionrequired by the energy minimization process.In order to choose the initial value required for the iterativeoptimization process, it is also possible to take the same set ofnode positions of the non-deformed CAD mesh. In this case, theinitial value is the CAD mesh itself except for the control points.The result obtained using this method is not acceptable comparedto the one obtained when selecting the initial value from the RBFinterpolation. In Fig. 9(a), the orange curve represents the non-deformed CAD mesh. The initial mesh from the CAD is shown inblue and the deformed mesh obtained applying the spring modelis shown as a red curve. As the initial value is taken from the CAD,the blue curve overlaps the orange one except in the control points.Fig. 9(b) shows a comparison between the two results: the resultobtained with the initialization by RBFs is shown in blue and theresult of the initialization with the position of the non-deformedis shown in red. The green curve is very close to the blue one andrepresents the FEM deformation used as a reference. 4.2.2. Energy optimizationThis process determines the value of the nodes’ final positionson the reduced mesh qj that present the minimum value in theenergy function ET defined by Eq. (10). This is an iterative processwhere given an initial position, a new position that minimizethe system’s energy function is calculated at each iteration. Asdescribed previously, RBFs are applied in order to generate theinitial value for the iterative process. The required value is takenfrom the reduced deformed mesh by using interpolation with theRBFs.In order to compute the minimization process, an iterativesolution function fminunc fromMatlab R ⃝ is used [33]. This functionis based on an algorithm using the well known quasi-Newtonmethod. In this method, using the gradient of the function asa starting point, the information of the curve is built in everyiteration to structure a quadratic problem as follows:minx12xTHx + cTx + b, (16)where is the Hessian matrix H and is assumed to be a positivedefinite matrix, c is a constant vector, and b is a constant. Thesolution to this problem is found when the partial derivatives ofx converge towards zero, that is:∇f (x∗) = Hx∗+ c. (17)The value for the optimal solution x∗ is expressed by:x∗= −H−1c. (18)In this implementation, x represents the positions of the nodesof the deformed and simplified model.4.3. Final interpolation using RBFsThe final step at interpolating the full model deformation usingRBFs. This time the nodes of the reducedmesh are taken as controlpoints and the interpolation is applied on all the nodes of theoriginal CAD mesh (Fig. 10). The result obtained in this processis the full and deformed polygonal mesh which represents anapproximation of the real deformed model.5. Experimental results and analysisIn order to validate the method based on the proposedspring–mass model, a Matlab R ⃝ implementation was developed.Tests were carried out in a machine equipped with an Intel CoreDuo with a 2.16 GHz processor, 2.0 GB in RAM, and Microsoft Windows XP. Data from the real deformed parts were acquiredusing a range scanner Vivid 9i from Minolta. Fig. 11 shows theexperimental setup. The calculation of the reference deformationfor the synthetic part was performed using SAP2000 R ⃝ package.This section presents the results of the tests carried out on asynthetic model and the models of three real parts. For each ofthe parts, a comparison of the deformation computed with theproposed method was compared against a reference deformation.For the synthetic part the reference model is obtained by using anFEM simulation, whereas for the real parts the reference model isthe polygonal model built from the surface data of the deformedpart. The following are thematerial parameters of themodels usedin the FEM simulations: Young module = 25.000 kgf/cm2andPoisson ratio = 0.35.The geometrical dimensions of the bounding box on thecoordinates (x, y, z) and the average thickness h of the parts areindicated in Table 1. The Table 2 shows numerical values of thedisplacement applied to the fixation nodes of the parts.Figs. 12(a), 13(a), 14(a), and 15(a) show a visual comparisonbetween the non-deformed CAD model (in orange), the deformedmodel used as reference (in green) and the deformed model usingthe spring system (in blue). The transversal cut shown in thosefigures is the one generated by section in Figs. 12(c) 13(c), 14(c),and 15(c). Figs. 12(b), 13(b), 14(b), and 15(b) show the reducedpolygonal model used to calculate the deformation. The blackpoints are the fixation points and the arrows indicate the directionof the applied displacements,whose numerical values are shown in Table 2. Figs. 12(d) 13(d), 14(d), and 15(d) represent the deviationsof the deformedmodel obtainedwith the spring systemin relationto the reference deformation.In all cases, the deformed model that results from theapplication of the proposed method is similar to the referencedeformation. Nevertheless, whereas in models 1 and 2 thedeviations are smaller than the average thickness of the parts,in models 3 and 4 there are regions with deviations greater. Inthe tests, some factors that affect the quality of the deformationapproximation have been identified. A factor that, in general,affects the result of the simulations is the irregularity of the partthickness, whereas the synthetic part is defined with uniformthickness and the part 2 presents high uniformity in thickness,parts 3 and 4 present noticeable variations in their thickness.Similarly, the complexity of the part and the degree of themesh reduction are contributing factors to the precision ofthe simulation.