A Taylor Expansion Approach For Computing Structural Performance .
A Taylor expansion approach for computingstructural performance variation frompopulation-based shape data Xilu Wang,Xiaoping Qian†Computational Design & Manufacturing LaboratoryDepartment of Mechanical EngineeringUniversity of Wisconsin-MadisonMadison, Wisconsin 53705Email: qian@engr.wisc.eduRapid advancement of sensor technologies and computingpower has led to wide availability of massive populationbased shape data. In this paper, we present a Taylor expansion based method for computing structural performancevariation over its shape population. The proposed methodtakes four steps: 1) learning the shape parameters and theirprobabilistic distributions through the statistical shape modeling; 2) deriving analytical sensitivity of structural performance over shape parameter; 3) approximating the explicitfunction relationship between the FE solution and the shapeparameters through Taylor expansion; 4) computing the performance variation by the explicit function relationship.To overcome the potential inaccuracy of Taylor expansion for highly nonlinear problems, a multi-point Taylor expansion technique is proposed, where the parameter space ispartitioned into different regions and multiple Taylor expansions are locally conducted. It works especially well whencombined with the dimensional reduction of the principalcomponent analysis in the statistical shape modeling.Numerical studies illustrates the accuracy and efficiencyof this method.Keywords: shape learning, statistical shape modeling, finiteelement analysis1INTRODUCTIONRapid advancement of sensor miniaturization and growing sensor network and computer power have lead to wideavailability of massive shape data from populations of objects. Such massive shape data range from human bodyshapes in Civilian American and European Surface Anthropometry Resource [1] and Size China [2], to longitudinalknee observations of a large population of osteoarthritis patients [3]. Populations of shape data also include shapesof man-made objects, such as shapes of same manufactured Anearlier version of this paper appeared in IDETC 2016author.† Correspondingparts due to manufacturing process variation as well as partshapes due to shape degradation after deployment. Miningand analysis of such massive population-based shape datacan result in knowledge of shape variability of the populationand lead to construction of faithful subject-specific 3D shapemodels from sparse measurements. It is then possible to predict shape-specific functional performance and populationwide structural performance variation. Such an ability bringsabout unprecedented capabilities and tantalizing opportunities for mass customization, part-specific failure prediction and just-in-time part maintenance, and patient-specificbiomedical intervention and treatment [4,5]. The goal of thiswork is thus to develop an efficient computer method that canpredict 1) shape-specific structural performance from givendiscrete shape data, and 2) structural performance variationover the shape population.Our approach builds on statistical analysis of shape variations, a.k.a. statistcal shape modeling (SSM). SSM hasemerged as a powerful tool [6] for shape learning froma population where statistical analysis of shape variationis conducted, typically through principal component analysis. It has found its success in various fields including image segmentation [7], motion tracking [8], and parametricshape design [9, 10]. The use of statistical shape modeling techniques to understand shape variations and its effecton biomechanical performance has been recently attemptedin [4,11,12,12–14]. However, the computing of the structureperformance variation over a population is usually throughthe Monte Carlo simulation, i.e. by randomly generating theshape parameters according to the learned probabilistic distributions, and obtaining a set of new shapes and new finiteelement meshes usually through mesh deformation. The finite element analysis is then performed on each of the generated finite element models and the results are collected fromwhich the structural performance variation is obtained. Forexample, in [14], the performance of the cementless osseointegrated tibial tray in a general population was studied using
capture a majority of the shape variations. Thus the regionpartition is only carried with respect to the first few shapeparameters and the dimension of the problem has been significantly reduced. To determining the positions of the expansion bases, an optimization based approach is designed.(a) A shape population2(b) FE mesh of the mean shape(c) FE result of (b)METHOD OVERVIEWThe proposed method involves four steps: 1) statisticalshape modeling; 2) shape sensitivity analysis; 3) approximation by Taylor expansion; and 4) computing the performancevariation.n The input for oour method are the boundary shapes:(k)XΓ , k 1, · · · , ns . Firstly, the shape parameters and theirprobabilistic distributions are learned from the statisticalshape modeling. The shapes are parameterized by the eigenψk } and the shape parameters {wk }shapes {ψm(d) Subject-specific temperature distributions from Taylor expansionXΓ XΓ wk ψ k , wk N(0, λk ),(1)k 1Fig. 1. Proposed approach for predicting subject-specific structuralperformance: Taylor series expansion of the FE solution for the meanshape as applied in a heat transfer problem.1000 finite element analyses on different subjects. The drawback of such Monte Carlo simulation based approach lies inits inefficiency. In order to obtain a result with reasonableaccuracy, a large number of experiments (usually 500) areneeded and each experiment, in this context, requires an expensive finite element analysis.The advantage of our approach lies in the fact that onlyone time FE analysis of the mean shape is used to predictthe subject-specific shape’s structural performance and theperformance variation over the population. Figure 1 showsthe proposed approach as applied in a heat transfer problem: approximating temperature fields for a shape population with Taylor expansion of the FE solution for the meanshape. Given a population of shapes, the mean shape andthe modes of variation are obtained by the statistical shapemodeling. A shape instance is then represented as the linearcombination of the mean shape and the modes of variation.The weights w [w1 · · · wm ]T is called the shape parameters.Based on the shape sensitivity analysis u/ w, where u isthe FE solution, the Taylor expansion is conducted to approximate the function u(w) and extrapolate the solution onthe mean shape to other shapes.The limitation of usual Taylor expansion lies in its potential inaccuracy for nonlinear problems. In this work, toovercome potential inaccuracy of Taylor expansion, a multipoint based Taylor expansion technique is proposed. The parameter space of the shape population is partitioned into different regions and multiple Taylor expansions are conductedaround the local bases within each region. This techniqueis extremely powerful when combined with the dimensionalreduction of principal component analysis (PCA) in SSM.Since as a result of PCA, the first several shape parameterswhere XΓ is the boundary shape, XΓ is mean shape of thepopulation. wk N(0, λk ) means the kth shape parameter isnormally distribution with mean 0 and variance λk .Secondly, the analytical sensitivities of the FE solutionu and the structural performance c with respect to the shapeparameters w [w1 , · · · , wm ]T are calculated c c u t, k 1, · · · , m. wk u wk(2)Then, with the results of sensitivity analysis, an explicitfunction relationship between the structural performance andthe shape parameters is obtained from the Taylor expansionm cwk , wkk 1c̃(w) c(0) (3)where c(0) is the structural performance of the mean shape.c̃(w) is the approximation of c(w).Finally, the probability distribution p(c̃) of the structuralperformance is obtained by the explicit function relationshipc̃(w) and the learned probability distribution of the shape parameters p(w).The output is the cumulative distribution function (CDF)Fc̃ (c ) p(c̃ c ) of the structural performance c̃, wherep(c̃ c ) is the probability that c̃ is less than c .In this study, we use p(·) to denote the probability ofa specific event. The boundary shapes are represented bydiscrete sampling points: XΓ [x1 , · · · , xnb ]t , where xi [xi,1 , xi,2 ]t is vector of coordinates and nb is the number ofsampling points. We use X to denote the FE mesh in general. When X appears in the equations, it means the positionsof the mesh nodes, for example, X [p1 , · · · , pn p ]t , where
pi [pi,1 , pi,2 ]t is vector of coordinates and n p is the numberof mesh nodes. We use u to denote the FE solution in general. When u appears in the equations, it means the nodalvalues of the FE solution, for example, u [u1 , · · · , un p ]t . If uu is an n p 1 vector and w is an m 1 vector, wt would bean n p m matrix, its element in the ith row and jth column uiis w.jSTATISTICAL SHAPE MODELINGStatistical shape modeling plays important roles in computing the structural performance variation over a shape population. It computes the mean shape and the modes of variations in a population. It captures the variability of shapesin space through the probability distribution of the learnedshape parameters. The mean shape of the population provides a statistical atlas, based on which we create the template FE mesh. The statistical shape modeling usually contains three steps: shape registration, shape alignment, andshape .02-0.03-0.0200.020.04-0.02 -0.01(a) Aligned training shapes00.010.020.03(b) Mean shape3(1)(n )sGiven a population of shapes Xraw , · · · , Xraw, theycan be registered to a reference shape by the rigid [6] or thenon-rigid shape registration techniques [15, 16], or could bycross-parameterization [17]. In this work, the shape registration is conducted by deforming the given shapes to the reference shape through the iterative free-form deformation [16],by which the boundary correspondences between the reference shape and the given shapes are obtained. Then, theshapes are re-sampled by the same number of points in correspondence and are aligned to the same coordinate frame,for example, by generalized Procrustes analysis [18]. These(k)(k)(k)re-sampled shapes are XΓ [x1 , · · · , xnb ]t , k 1, · · · , ns ,where ns is the number of training shapes, and nb is the number of sampling points on each shape.The principal component analysis (PCA) is then conducted for capturing shape variations and dimensional reduction. In PCA, each shape is treated as a vector in R2nb , themean shape of the population isXΓ 1 ns (k) XΓ ,ns k 1(4)the shape covariance matrix isΣ (1)1ΦΦT ,ns 1(5)(n )where Φ [XΓ XΓ , · · · , XΓ s XΓ ]. Just as all the covariance matrices do, the shape covariance matrix Σ describesthe patterns and ranges of variations from the mean XΓ .Through eigen-decomposition, we haveΣ ψ k λk ψ k , k 1, · · · , ns 1,(6)(c) First modeFig. 2.(d) Second mode(e) Third modeStatistical shape modeling of hand shapes: (a) 40 alignedtraining shapes; (b) the mean shape: XΓ ; (c) the first mode byvarying w1 : XΓ w1 ψ 1 ; (d) the second mode by varying w2 :XΓ w2 ψ 2 ; (e) the third mode by varying w3 : XΓ w3 ψ 3 .where ψ 1 , · · · , ψ ns 1 are the principal components of Σ ,λ1 , · · · , λns 1 are the corresponding eigenvalues. The principal components captures the modes of shape variations.The eigenvalues are the amount of variances in those components.Through PCA, the shapes are modeled as the linear combination of the mean shape and the variation modesmXΓ XΓ wk ψ k ,(7)k 1where XΓ is a shape modeled by PCA, ψ k , k 1, · · · , m arethe first m variation modes. The number of modes, m, can bemns 1k 1k 1determined, e.g. from λk / λk 98%, which meansthat the first m modes should capture more than 98% of thetotal shape variances in the training set. wk , k 1, · · · , m arethe corresponding weights. We call them the shape parameters and note w [w1 , · · · , wm ]t .In Figure 2 we show an example of statistical shapemodeling of 40 hand shapes, from which we see the meanshape of the population and the first three modes of shapevariations. As a result of dimensionality reduction, the firstthree shape modes captures 66.6%, 16.6%, and 7.8% of thetotal shape variances respectively.The shape parameters {wk } are assumed to be normallydistributed. They are uncorrelated with each other as the result of PCA. The probabilistic distribution of the shapes inR2nb is then modeled by the distributions of the first m shapeparameters aswk N(0, σk ), k 1, · · · , m,where N(0, σk ) stands for the normalp distribution with 0mean and standard deviation σk λk .
4SHAPE SENSITIVITY ANALYSISThrough statistical shape modeling, a shape in the population is parameterized by the shape parameters w. In thissection we derive the analytical sensitivity of the FE solutionu over the shape parameters w. Assume we have the FE stateequationK(w)u b(w),(8)the sensitivity of the FE solution over the shape parametersis calculated by [19]nb 2 matrix of TPS weights with vi [vi,1 , vi,2 ]t . The kernelfunction ρ is defined as ρ(h) khk2 log(khk), khk 0;0,khk 0.Given the coordinates of the initial boundary pointsxΓ1 , · · · , xΓnb , and the coordinates of the perturbed boundarypoints xΓ1 , · · · , xΓnb , the translation vector c, affine transformation matrix A and the deformation weights V can be solvedas [6, 22]:11 u K 1 wk bb K u , k 1, · · · , m. wk wkV B XΓ ,(9)For a specific element e in the stiffness matrix K or theloading vector b, by the chain rule we have e X XΓ e , wk Xt XtΓ wk(10) ewhere Xt 1 2n p is the sensitivity of element e with respectto the mesh nodes, it is calculated according to the governing Xequations of the finite element method [19–21]; Xt 2n p 2nbΓis the sensitivity of the mesh nodes with respect to the boundary points, it is calculated based on the Thin-plate deformaΓtion, whose details will be given later; and X wk 2nb 1 is thesensitivity of the boundary points with respect to the shapeparameters and from equation (7) we have XΓ ψ k , k 1, · · · , m. wkThe only unknown in equation (10) now isHerethe thin-plate deformation (TPS) [22] is used to transfer theboundary perturbations to the interior nodes due to its simplicity and robustness. It is worth mentioning that other deformation methods could also be applied, for example, thefree-form deformation [23], the deformation by pseudo linear elasticity [24].The formulation of the thin-plate deformation isΦ (x) c Ax Vt U(x),Φ (x) XtΓ B12 ct B21 XΓ ,At(14) 1 XtΓ B11 U(x),x(15)where B12 is the transpose of B21 .Assume X [p1 , · · · , pn p ]t the nodes of the FE mesh ofthe mean shape XΓ , and X [p1 , · · · , pn p ]t the nodes of thedeformed FE mesh. We have X as a linear function of theboundary points XΓ tX [1, X ]B21 Ut (X)B11 XΓ ,(16)where U(X) [U(p1 ), · · · , U(pn p )]. From equation (16) wehave the sensitivity of the mesh nodes with respect to theboundary points X XtΓ!t[1, X ]B21 Ut (X)B11t[1, X ]B21 Ut (X)B11,(17)on the left side of the equation X and XΓ are vectorized.Now, we have the sensitivity of the FE nodal solutions ufromequations (9) and (10). The sensitivity of the structw ctural performance wt can be easily obtained by the chainrule:(12)where x [x1 , x2 ]t is the domain point and is deformed toΦ (x); c [c1 , c2 ]t is the translation vector; A is the 2 2affine transformation matrix; Vt U(x) is the deformation part,where U(x) [ρ(x xΓ1 ), · · · , ρ(x xΓnb )]t is the nb 1 vectorof kernel functions, xΓ1 , · · · , xΓnb are the coordinates of boundary points with xΓi [xΓi,1 , xΓi,2 ]t , and V [v1 , · · · , vnb ]t is the where XΓ [xΓ1 , · · · , xΓnb ]t is the nb 2 matrix of initialboundary points, XΓ [xΓ1 , · · · , xΓnb ]t is the nb 2 matrix ofperturbed boundary points. In this paper the mean shape XΓis set as the initial boundary. B11 (nb nb ) and B21 (3 nb )are coefficient matrices decided by the positions of the initialboundary points XΓ , whose closed form formulation is givenin [6].Substituting (14) into (12), we have(11) X. XtΓ(13) c c u t. wt u wt5(18)TAYLOR APPROXIMATION OF STRUCTURALPERFORMANCEThe Taylor expansion is used to explicitly approximatethe function relationship c(w) between the structural performance c and the shape parameters w.
1ww1w1 w1Tc c(p1) c/ wT (w-p1)pp1 1 c c(p1) c/ w (w-p1)pp0 0pp2 2p1 p1p3 p3ww2 2T(w-p )T(w-pc c(pc c(p0) c/ w0) c/ w0) 0p0 p0 p4 p4w2 w2p2 p2j 0,··· ,nT(w-p )T(w-pc c(pc c(p2) c/ w2) c/ w2) 2(a) Three regions(b) Five regionsFig. 3. Partition the domain into different regions and conduct Taylorexpansion in each region separately.5.1Single point Taylor expansionIn the single point Taylor expansion, the performancefunction c(w) is expanded around the mean shape, where theshape parameters are zeros.c̃(w) c(0) cw, wtIt should be noted that, though c̃(w) may not be continuous,the obtained probability distribution p(c̃) will be close to thatof the true performance p(c) as long as c̃(w) is close to c(w).Choosing appropriate expansion bases p0 · · · pn is critical in the multi-point Taylor expansion. For each point w,it is expanded with respect to the closest base point. So therange of extrapolation at w is l(w) min kw p j k. Since(19)where c(0) c(u(0)), and u(0) is the FE solution on themean shape solved from the below equation:the error of Taylor expansion is propositional to l(w)r 1 ,where r is the degree of expansion and in this paper r 1, itis desirable to minimize the overall squared range of extrapolation. However, each shape parameter w does not appear inthe same frequency, the accuracy of approximation is moreimportant at the regions of high probability. Based on that,an objective function is designed:Zmin E {p j }l(w)2 p(w)dw,(22)Ωwhere the extrapolation range l(w)2 is weighted by the probability density p(w) and is integrated over the whole domain.Note that l(w) in equation (22) is a minimum formula andwill cause obstacles for the optimization. Since the p-normis widely used in approximating the minimum and maximum formulas, l 2 (w) min kw p j k2 is substituted byj 0,··· ,nK(0)u b(0).(20)5.2Multi-point Taylor expansionTo overcome the potential inaccuracy of Taylor expansion at points far away from the mean shape, a multi-pointTaylor expansion technique is proposed.As shown in Figure 3 are two different examples of themulti-point Taylor expansion. In Figure 3(a) the domain hasbeen partitioned into three regions by the two dashed horizontal lines. The Taylor expansions are conducted locally ineach region around the bases pi , i 0, 1, 2. In Figure 3(b) thedomain has been partitioned into five regions. The procedureof the multi-region Taylor expansion is as follows:1. Choosing n number of expansion bases {p0 , p1 , · · · , pn }.2. Partition the parametric domain into n regionsΩ0 , Ω1 , · · · , Ωn according to the closest distance tothe base points: Ωi w kw pi k min kw p j k .j 0,··· ,n3. Approximate c(w) piece-wisely by the Taylor expansions around the local bases: 2q1( ni 0 l j ) q . Considered that the probability density p(w)of the shape parameters is original symmetric as in equation(24), it is desirable to have the expansion bases symmetricwith respect to the origin. At last, we have the optimizationformula:Zmin E {p j }n 2q q1)( l jΩp(w)dw,(23)i 0s.t. p j pn j , j 0, · · · , n.Locations of expansion bases are then obtained from (23)through a gradient decent approach.6Probabilistic distribution of the structural performanceThrough the Taylor expansion, an explicit function relationship c̃(w) between the shape parameters w and the structural performance c is obtained. By the statistical shape modeling, the probability density function of the shape parameters are learned:w2 c c(p0 ) wT (w p0 ), w Ω0.c̃(w) . cc(pn ) wT (w pn ), w Ωnk 12 2λkp(w) Πmek 1 (2πλk )(21),(24)where wk is the kth shape parameter and λk is the shape variance in the kth principal direction.
The cumulative distribution function of the approximated structural performance c̃ is given by:Fc̃ (c ) p(c̃ c ) Zp(w)dw.(25)c̃(w) c 6.1Closed form solutionIf the obtained structural performance c̃ is linear andcontinuous as in equation (19), since it is assumed that theshape parameters are normally distributed, we have that c̃ isalso normally distributed with mean c(0) and variance:(a) Boundary conditions on themean shapeFig. 4.(b) FE solutionA 2D heat transfer problem: Dirichlet boundary condition u u 50 on Γ1 (red circle), Neumann boundary condition n 200(green boundary), thermal load: q 1000000 in the centerof the hand (yellow area).λc c cΛ , wt w(26)where Λ diag(λ1 · · · λm ) is the covariance matrix of w. Theclosed form of equation (25) is then obtained accordingly.which captures more than 98% of the total shape variance.The template FE mesh is created on the mean shape and hasn p 1250 number of nodes.7.16.2Monte Carlo integrationIf the obtained structural performance c̃ is discontinuous as in equation (21). A closed form of equation (25) iseither non-existent or very hard to obtain. In such cases, theMonte Carlo integration [25] is used to integrate the cumulative probability function (25). Compared with the finite element analysis, the function evaluations of c̃(w) by (21) costnothing.7NUMERICAL RESULTSIn this section, the influence of geometrical variation onthe structural performances have been studied with a 2D heattransfer problem and a 2D elasticity problem. We examinethe numerical accuracy of the Taylor expansion for variousmodes of shape variations. We also compare the distributionsof the structural performances obtained by Taylor expansionwith those obtained by the Monte Carlo simulation.The evaluation process of Monte Carlo simulation(MCS) is as follows:1. Randomly generate N 500 sets of shape parameters{wi } according to p(w).2. For each shape parameter wi , generate the correspond(i)ing boundary shape XΓ by the statistical shape model(7).(i)3. Generate the finite element mesh X(i) for XΓ by thethin-plate deformation of the FE mesh of the mean shape(16).4. Conduct the FE analysis, record the results and repeatsteps 2,3,4 until i N.In our numerical study, the 40 hand shapes in [26] areused as the training set as in Figure 2. Each shape is represented by nb 2001 number of discrete points. We modelthe shape variations among them through the statistical shapemodeling method as detailed in [16]. The first 8 shape modesis used to compactly represent the overall shape variation,2D heat transfer problemThe 2D heat transfer is governed by the Poisson equation: u q in Ω(27)u T1 on Γ1 u g on Γ2 n(28)(29)where u is the temperature, q is the thermal load, Ω is thedomain of heat transfer, Γ1 is the Dirichlet boundary, T1 isthe boundary temperature, Γ2 is the Neumann boundary, andg is the Neumann boundary condition.Figure 4 shows the 2D heat transfer on the hand shape.Figure 4(a) shows the FE mesh of the mean shape and theboundary conditions. Figure 4(b) shows the correspondingFE solution. In Rthis example, the variability of the thermalcompliance c Ω qudΩ due to the shape variations is studied.7.1.1 Temperature distribution by Taylor expansionFigure 5 shows the predicted temperature distribution ofshapes due to the change of the first shape parameter. Wecan see that as w1 increases from 2σ1 to 2σ1 in Figure 5(a),5(b), 5(c), and 5(d), the hand becomes more expanded, andthe temperature in the field decreases.In order to examine the accuracy of Taylor expansion,here we compare the results predicted by the Taylor expansion with the ones obtained by the finite element analysis.The Taylor expansion of the nodal temperatures u(w)and the thermal compliance c(w) around the mean are donethrough equation (19). The finite element analysis are conducted at the designed points, as shown by the stars in Figure 7. At each point, a new shape is generated by the corresponding shape parameters, the FE mesh is obtained by thethin-plate deformation of the FE mesh of the mean shape anda new FE analysis is conducted.
#109#1097.5FE SolutionTaylor expan.76.8FE SolutionTaylor expan.6.6cc6.56.465.5e(w1 ) w1 2σ1(a) ue(w1 ) w1 σ1(b) u6.2-3-2-10123-3-2-1w1/ 1(a) First modeCumulative ProbabilityFE SolutionTaylor expan.c6.6Fig. 5. Predicted temperature distribution due to shape variationsin the first mode. The color means the temperature, and its rangefollows the same color bar as in Figure 4.6.56.46.330.80.6Taylor expan.MCS0.50.40.20.056.2-320.956.7e(w1 ) w1 2σ1(d) u1(b) Second mode#109e(w1 ) w1 σ1(c) u0w2/ 2-2-101w3/ 3(c) Third mode235.566.5c77.5#109(d) CDFFig. 7. Comparing Taylor expansion with FE analysis of thermale c(0) w1 c/ w1 ; (b) ce c(0) w2 c/ w2 ;compliance: (a) ce c(0) w3 c/ w3 ; (d) cumulative distribution functions from(c) cTaylor expansions and from 500 Monte Carlo simulations.e(w1 ) u(w1 ) w1 2σ1(a) ue(w1 ) u(w1 ) w1 2σ1(b) ue(w2 ) u(w2 ) w2 2σ2(c) ue(w2 ) u(w2 ) w2 2σ2(d) ue(w3 ) u(w3 ) w3 2σ3(e) ue(w3 ) u(w3 ) w3 2σ3(f) uFig. 6. The errors between the temperatures predicted by Taylorexpansion and from FE analysis.Figure 6(a) and 6(b) show the errors between the temperatures predicted by Taylor expansion and from FE analysis with shape change in the first mode. The results areobtained by varying the first shape parameter from negativetwo standard deviations to positive two standard deviations,while keeping all the other shape parameters 0. The maximum errors in Figure 6(a) and 6(b) are 6.89 and 7.03, respectively, while the scale of temperature variation in our FEsolution is about 80 as shown in Figure 4. We could seethat the maximum error happens at the tip of the little finger,where the shape variation is large and is far from the Dirich-let boundary. The Taylor expansion extrapolates the FE solution of the mean shape to other shapes, so it is reasonable toexpect that the maximum error happens at the farthest extrapolation point (large shape deviation). Since the temperatureon the Dirichlet boundary is fixed, so there is no error on theDirichlet boundary.Figure 6(c) and 6(d) compare the Taylor expansion withthe FE solutions for shape changes along the second mode.The maximum errors in Figure 6(c) and 6(d) are 1.59 and2.15, respectively. Figure 6(e) and 6(f) compares the Taylorexpansion with the FE solutions for shape changes along thethird mode. The maximum errors in Figure 6(e) and 6(f) are0.80 and 0.81, respectively.It could be seen that from the first mode to the thirdmode, the errors become smaller and smaller. That’s because as a result of PCA, the first mode captures a majority ofthe total shape variances and the remaining modes capturesfewer and fewer shape variances. So the deviation from themean shape becomes smaller and smaller.7.1.2 Thermal compliance by Taylor expansionFigure 7(a) shows the relationship between the firstshape parameter w1 and the thermal compliance c by Taylor expansion and from the FE analysis. Since the thermalload q as in Figure 4(a) is added in the middle area, wherehas small extrapolation errors as in Figure 7(a), the results ofTaylor expansion c̃ is close to the FE analysis c. The maximum relative error (max ec c /c) is 6.77%.Figure 7(b) and 7(c) show the relationships between thesecond and third shape parameters with the thermal compliance. The results of Taylor expansion agree well with the finite element analysis, the maximum relative errors are 0.42%and 0.49% respectively.Since the Taylor expansions in Figure 7(a), 7(b), and
e(w1 ) w1 σ1(b) ue(w1 ) w1 σ1(c) ue(w1 ) w1 2σ1(d) u(b) Nodal displacements0.15FE SolutionTaylor expan.c0.10.06-20 · σ f in Ω, · u)I in Ω,σ 2µεε λ( 1 u uT ) in Ω,ε ( 2u û on ΓD ,σ n t̂ on ΓN ,(30)(31)(32)(33)(34)where σ is the domain stress, f is the domain force, Ω is thedomain, u is the displacement, I is the identity matrix, û isthe fixed displacement on the Dirichlet boundary ΓD , and t̂is the traction on the Neumann boundary ΓN .In thisexample, the variability of the structural compliRance c ΓN uT t̂dΓ due to the shape variations is studied.Figure 8(a) shows the FE model X of the mean shape XΓand the boundary conditions. Figure 8(b) shows the solvednodal displacements.2-20w1/ 10.075(b) Second modeFE SolutionTaylor expan.0.070.065-202w2/ 2(a) First mode(c) Third mode2D elasticity problemThe governing PDEs of the linear elasticity problem are0.080.070.05w3/ 37.2FE SolutionTaylor expan.0.09Cumulative Probability7(c) gives relatively small extrapolation errors, the singlepoint Taylor expansion (19) around the mean shape is usedto approximate c(w) and calculate the cumulative probability function (CDF) of the thermal compliance. In this casewe have the analytical solution (26). The result is shown inFigure 7(d), it can be seen that the analytical CDF conformswell to that obtained by the Monte Carlo simulations. Thethree horizontal curves in Figure 7(d) partition the space intofour intervals at the cumulative probabilities of 5%, 50%,and 95%. From the two inner intervals we could see that, forabout 90% of the shapes in the population, the thermal compliance c should be within the range [5.8 109 , 7.1 109 ].The run time for the Taylor expansion based approach is2.88s including the sensitivity calculation. The run time forthe 500 Monte Carlo simulations is 106.54s. The computingis performed with MATLAB on the processor of “intel(R)Core(TM) i7-5500U”.Fig. 9. Taylor expansion predicted nodal displacements due toshape variations in the first mode. The color shows the values ofhorizontal displacements and its range follows the same color bar asin Figu
work is thus to develop an efficient computer method that can predict 1) shape-specific structural performance from given discrete shape data, and 2) structural performance variation over the shape population. Our approach builds on statistical analysis of shape vari-ations, a.k.a. statistcal shape modeling (SSM). SSM has
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Hotell För hotell anges de tre klasserna A/B, C och D. Det betyder att den "normala" standarden C är acceptabel men att motiven för en högre standard är starka. Ljudklass C motsvarar de tidigare normkraven för hotell, ljudklass A/B motsvarar kraven för moderna hotell med hög standard och ljudklass D kan användas vid
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12.2 Thermal Expansion Most materials expand when heated and contract when cooled. Thermal expansion is a consequence of the change in the dimensions of a body accompanying a change in temperature. 3 types of expansion: Linear expansion. area expansion, volume expansion In solid, all types of thermal expansion are occurred.
Taylor series As the degree of the Taylor polynomial rises, it approaches the correct function. This image shows and Taylor approximations, polynomials of degree 1, 3, 5, 7, 9, 11 and 13. The exponential function (in blue), and the sum of the first n 1 terms of its Taylor series at 0 (in red).
