## 2. Methodology of the Parametric Model

The methodology of the parametric computation regarding the structural problems contains four main steps in total: the measurement, the parametric approximation, the parametric modelling, and the computation, as is shown in

Figure 5. The common computation method constructs the target object as a simplified form in the CAD modelling software, which only keeps basic features regarding some sizes, while the deformed details are omitted. Measurement experiments with accurate sensors are desired, which aims to reconstruct the deformed information. Measured data is converted to the available parameter set and then extracted into different features. The 3D model can be fitted by the B-spline volume directly if the model shape is not complex [

28], which is not common in structural problems. Therefore, volume features are further decomposed into surface patches. The B-spline surface approximation is applied to fit the decomposed surfaces. The final parametric surfaces are only acceptable when the fitting accuracy is satisfactory. The parametric surfaces are converted to the CAD modelling software. The parametric geometric modelling is processed based on the parametric surfaces and transformed into the computational model. Lastly, the computation progress can be accomplished after the mesh generation and boundary condition application. Kernel links are listed in the next sections in details and applied into a composite structural building.

#### 2.1. Feature Acquisition of the Object

In the feature acquisition progress, it is initially necessary to obtain the global feature of the target object through measurement methods by advance sensors. A TLS-based sensor system is adopted in this step. Secondly, the task is to transform the large datasets into the key and useful coordinate values. Thirdly, the patch decomposition is carried out due to the discontinuity characteristics in the right angle parts, which is ubiquitous for complex building and mechanical structures. There are also some windows, entrance and exit passages, and non-removable parts on the surfaces. Finally, as a result, it is necessary to recognize the boundary features that represent windows, passages, and non-removable parts. In the following, the detailed experiment and operations are introduced to obtain the object feature with the application of these steps.

The researched object is used on the basis of a composite structure regarding a warehouse building with a north-wall length of about 16 m, south-wall length of about 16.5 m, width of about 14.2 m, height of 3.6 m, and wall thickness of 0.2 m. The point cloud of this structure is sample data from PointCab GmbH which offers powerful software to make the processing of high-resolution point clouds easy. It also offers open access for users to download sample data regarding point clouds. The aim is to research weak situations and positions in this deformed structure by combining TLS with the FE computation. The TLS equipment (FARO Focus

^{S} 350) and laser tracker equipment (FARO Laser Tracker Xi) were used here to obtain the 3D point clouds [

46,

61,

62,

63]. The systematic distance error of this type TLS is

$\pm 1\mathrm{mm}$. The vertical and horizontal resolution of the TLS is

$0.009\xb0$. The maximum vertical scanning speed of the TLS is

$97\mathrm{Hz}$. The laser tracker, which has an angular accuracy of

$18\mathsf{\mu}\mathrm{m}+3\mathsf{\mu}\mathrm{m}/\mathrm{m}$ in angle measurement performance, is applied for the purpose of reference and validation.

The scanning positions are separated into three parts, as it is shown in

Figure 6 from the top view. The north direction is marked as N here. There is a total of 13 positions for scanning totally. From the ground-based scanning positions outside the building, there are five positions to capture the outside details: 1, 2, 3, 4, and 5. There are also five scanning positions inside the building: 6, 7, 8, 9, and 10. There are normally only one or two scanning positions inside such a building. However, there are four forced pillars inside the building. Consequently, the number of scanning positions is increased to five to capture the details of the forced pillar. The final scanning positions 11, 12, and 13 are based on the building roof.

The point clouds measured from PointCab contain not only the data needed, but also the unexpected noisy points. A smoothing operation is commonly applied when the initial images are dealt with. This is known as blurring. The role of the smoothing operation here is to reduce the noisy points in point clouds from the TLS. The Gaussian Filter applying the OpenCV function “GaussianBlur” is adopted in this research. The size value and application of the function can be referred to in [

64]. The detailed structure of this warehouse building can be found in the point clouds of

Figure 7 after the smoothing operation. There are coordinate, scalar, RGB color, and normal values in the scanning results. The kernel information in

Figure 7 is the coordinate value of every point which is finally extracted as a P matrix by deleting the unexpected values in MATLAB R2018b, see Equation (1).

The focus of this research lies in the reconstruction of the complex composite structures with unknown deformations based on the parametric solution. Hence, points on the volume are firstly decomposed into points on the surface patches. The parametric solid volume is assembled again by these patches after accomplishing the appropriate surface approximations. This is the solution to reconstruct the parametric solid model in this research. The decomposed patches are shown in

Figure 8.

As shown in

Figure 8, the global volume is decomposed into five parts which are marked as east, west, south, north, and pillar. The door, window, and lower window features are represented as D1 to D5, W1 to W20, and L1 to L6, respectively. The following fitting process is based on the continuous B-spline surfaces. Consequently, surfaces connected with right-angle interface layer should be subdivided into smaller patches. The east outside wall in

Figure 9, which is implemented in the open access software CloudCompare V2.8, is subdivided into five patches by blue dotted layers. The purpose is to guarantee the continuity of every independent patch. As a result, the subdivision is adopted in other parts in

Figure 8, which is followed by the same method in

Figure 9. The boundary extractions of the window and door features apply the boundary threshold method [

6] using the principle of Delaunay triangulation [

64,

65] with a threshold value of 10 mm. The aim of this extraction is to recognize the boundary features of different surfaces for the future trimming step in the B-spline surface.

#### 2.2. B-Spline Approximation

#### 2.2.1. B-Spline Basis Function

Given a knot vector

$\Xi $, the B-spline basis function

${\left\{{\mathrm{N}}_{i,p}\right\}}_{i=1}^{n}$ starts with a zeroth order basis function (

$p=0$) as is shown in Equation (2). It is known as the Cox–deBoor recursion function [

66]. The values of

${\xi}_{i}$ are elements of knot vectors. Here, the parameter

$\xi $ is the coordinate in the parametric space that is needed to satisfy the relationship in Equation (2). The parameter

$\xi $ varies from the minimum value to the maximum value along the calculated curve.

The B-spline basis functions

${\left\{{\mathrm{N}}_{i,p}\right\}}_{i=1}^{n}$ are defined recursively by Equation (3), for

$p\ge 1$.

$p$ is the degree of the basic function. Here, it is possible for the status during the molecular formula computation in Equation (3) to behave in special situation as

$0/0$in the ratio of the numerator and the denominator. The convention of this special situation is to default this as 0 in the recursion work. For detailed explanations, please refer to [

12,

67].

#### 2.2.2. B-Spline Curves and Surfaces

The last remaining step is to create the control points

${Q}_{i}$ of the B-spline. They form the so-called control polygon in the calculation. Interested readers can refer to [

12,

13,

63] for more details regarding the B-spline parameter calculation.

The B-spline curve

$C\left(\xi \right)$ can be constructed by linearizing the B-spline basis functions, as is given by Equation (4). Coefficients of the basis functions are referred to as control points.

The B-spline surface${\mathit{S}}^{\mathit{B}}$ is represented as Equation (5), which is a tensor product of B-spline curves. The control points of the B-spline surface form a control net${Q}_{i,j}$,$i=0,1,\cdots ,n$,$j=0,1,\cdots ,m.$ ${N}_{i,p}\left(\xi \right)$ and${M}_{j,q}\left(\eta \right)$ are basis functions of B-spline curves which approximate B-spline curves in two directions.

#### 2.2.3. B-Spline Approximation

The first B-spline approximation task is based on curves which are of significance to describe different feature boundaries of the structure, for example, windows and doors. The degree value of basis functions is 3 [

6,

67]. The values of the control point number of different feature boundaries regarding the east side wall in

Figure 9 are listed in

Table 1. Four feature regions have to be extracted on the east outside surface. Ground-based sides of each feature are ignored. Other sides obtain reasonable numbers of controlled points as is shown in

Table 1, which are referred to in detail in [

12,

63]. The extracted B-spline feature curves are then applied as the trimming and projection boundaries to the B-spline surfaces.

Aiming at reconstructing the solid parametric model of composite structures, the surface-based approximation is a key step in the reconstruction step. Taking the south side surface as an example, the deformed surface is approximated as a parametric surface with the application of B-splines, as is shown in

Figure 10a. The degree value of the basis functions is 3. The degree value can be adjusted according to the approximation demands. The choice in this manuscript is based on previous papers [

6,

63]. The control points are distributed in two directions in the surface fitting process, as mentioned in Equation (5). In the south surface example, numbers of control points are 36 in the height direction and 165 in the length direction. The accuracy characteristic of B-spline surfaces in [

6,

63,

67] is proved which adopts 20 control points in the approximation progress regarding an arch structure with the length of 2 m, which indicated that there is a control point every 0.1 m. The determination regarding the quantity of control points affects the quality of the approximated surface. Detailed explanations regarding the parameter and model selection problem are given in [

63]. The length of this surface is about 16.5 m. The height is about 3.6 m. Hence, control points are added to the model every 0.1 m in this example. The standard deviations are calculated in

Section 4.1, which indicates that the quality of the parametric model is more accurate than the simplified model. Consequently, the number of the control points can be considered as reasonable and reliable in this computation.

Parametric surface trimming is a complex activity [

68]. The shape of B-spline curves regarding the feature parts are not absolutely compatible with the shape of the B-spline surface due to the uncertainty of the boundary extraction progress. Hence, the surface trimming step is of great significance to obtain accurate boundary recognition regarding the feature parts in the solid model. The final B-spline surface with window features after trimming is shown in

Figure 10b.

## 4. Discussion and Analysis

This research is based on the model reconstruction of the composite structures. The evaluation of different models is unknown. Consequently, it is necessary to validate the model quality and describe the main effects due to the model deviations in following subsections.

Section 4.1 discusses the quality of the reconstructed model itself and the deviation between the measured point cloud data and reconstructed models.

Section 4.2 analyzes the deviations between two researched models.

Section 4.3 extracts the deformation contour of both models into one figure and analyzes development laws regarding the deformation.

Section 4.4 carries out a detailed discussion regarding the stress result.

Section 4.5 analyzes the modal features of the dynamic computation.

#### 4.1. Model Quality Based on Measured Data

Model errors based on different surfaces are listed in detail in

Table 3 in order to determine the model qualities. The error analysis

${d}_{i}$ is based on the comparison between the point cloud matrix P and the analyzed model surfaces. The standard deviation

$\delta $ is calculated by Equation (6). In the parametric model based on B-splines, the standard deviations of all fitting surfaces are about or within 1 mm. The maximum standard deviation is 1.15 mm which is on the south outside surface. The maximum standard deviation in the simplified model is 22.56 mm which is on the north outside surface.

The maximum deviations in the parametric model are in the interval of [

26,

51] mm. These values are much larger than the standard deviations. This is due to the remaining noise points on these surfaces. Hence, the maximum deviation also partly contains outliers. The standard deviations of the east, south, and north surfaces are larger than that of other surfaces from

Table 3. It is also obvious from

Figure 8 that the boundary regions of door or window features are large. The possible reason for the larger standard deviation is that some outliers exist in large boundary regions in the TLS measurement. Both maximum and standard deviations in the simplified model are obvious due to neglecting of the surface deformation in the reconstruction progress of the simplified model.

Generally speaking, the approximation quality of the parametric model behaves superior to that of the simplified model on the basis of surface deviation comparisons, which is indicated in

Table 3. The average approximation accuracy of the parametric model can reach the millimeter level. The maximum deviations of all outside surfaces in the simplified model are dozens of times larger than the corresponding standard deviations in the parametric model. This indicates that the simplified model is far from the true measured data.

The standard deviation of the parametric model can still be reduced by increasing the control points and adjusting the order of basis functions in the approximation progress of the composite structures. However, the calculation cost and the workload of the model reconstruction will increase greatly. Meanwhile, the mesh generation and the FE computation will be more complex. As a result, the global computation of the parametric model will perform extremely inefficiently in this condition. In this manuscript, the object researched is a large scale model. It is important to balance the calculation cost and computation accuracy. The present accuracy of the parametric model is much higher than the accuracy of the common simplified model. The determination of the geometric accuracy differs in different scale simulations. The computational model reconstruction of details in composite structures with multi-scales can be improved by adjusting control points and basis function orders based on multi-scales.

#### 4.2. Deviation Analysis of Two Models

Details about the centroid position and moment of inertia regarding to different models are extracted in

Table 4. The mass volume refers to the size of the volume filled with model materials, which differs from the common solid enclosed volume. The mass volume of the simplified model is 67.8 m

^{3}, which is 88.6% of the mass volume of the parametric model. This indicates the weights of both models are different. As is generally known, the deformation exists when the force is loaded on the non-ideal rigid body. Therefore, the effects due to the different self-weights cause different predeformation effects. Hence, the computation based on the simplified model brings predeviation before the FE computation.

The centroid positions are listed in

Table 4. The local origin points of both models are set at the endpoint on the north lower side of the east outside surface. The local origin point coordinate in the global coordinate system of the simplified model is (0, 10.1, −9.9) m. The local origin point coordinate in the global coordinate system of the parametric model is (−21.2, −6.8, 0.2) m. The relative centroids in the global coordinate system can be calculated by Equation (7). In this equation,

$\mathit{R}\mathit{C}$ is the coordinate of the relative centroid,

$\mathit{C}$ is the coordinate of the centroid in global system, and

$\mathit{L}\mathit{O}\mathit{P}$ is the global coordinate of the local origin point of each model. The coordinates of each relative centroid are shown in

Table 4, respectively. The coordinate difference

$\left(\u2206x,\u2206y,\u2206z\right)$ between two relative centroids is (0.4, 0.5, 0.26) m, respectively. The deviation rate of the centroid coordinate based on the parametric model centroid is about (5.3%, 5.8%, 9.4%). As a result, the deviation rate of the moment of inertia in different directions is (4.8%, 6.7%, 6.1%). While the moment of inertia is greater, the ability to resist deformation of the object is stronger. Hence, the difference of the centroids and moments of inertia is also a possible reason for the deformation deviation after loading.

Surfaces in the simplified model are all based on flat planes. The surface

${\mathit{S}}^{s}$ of the simplified model constructed in this research is described by Equation (8) which is a plane. Hence, the distance of the B-spline surface point

${r}_{B}$ in the parametric model to the simplified flat surface

${\mathit{S}}^{s}$ is obtained by Equation (9).

${r}_{B}=\left({x}_{i},{y}_{i},{z}_{i}\right)\in {\mathit{S}}^{\mathit{B}}$. The total amount of approximated B-spline points is

$l$. The global average deviation

${D}_{\mathsf{\Sigma}}$ between the B-spline surface

${\mathit{S}}^{\mathit{B}}$ and the simplified surface

${\mathit{S}}^{s}$ can be indicated by Equation (10). Hence, the global average deviation

${D}_{\mathrm{average}}$ regarding 11 surfaces can be calculated by Equation (11). The final average deviation of the global surfaces is 8.76 mm. This value indicates the obvious shortcomings of the simplified model which is applied commonly. The parametric model describes the deformed details on all surfaces with the advantage of B-splines. However, the simplified model reconstructs the surfaces as idealized and flat ones and ignores the deformation information description. This is the main reason for this obvious deviation between both models.

In summary, it is of great significance to compare the deviation performance when the geometric models of structural problems are reconstructed before the FE computation. The deviations between the parametric and the simplified models are obvious. The mass volume of the parametric model is larger than that of the simplified model which results in the deviation of the predeformation due to the self-weight. The relative centroid of the parametric model offsets from the simplified model relative centroid in the global coordinate system. The average deviation between two models is finally very large, which leads to influence the parametric model in the model selection progress regarding the structural problems.

#### 4.3. Deformation Analysis Based on Static Structure

Aiming at researching the effect of different geometries on the FE computation, it is of significance to carry out the deformation analysis. The deformation contour figure regarding different deformation values of the roof surface is extracted in

Figure 15. The blue lines indicate deformation results based on the parametric model, while the red dashed lines show the deformation results based on the simplified model. The centroids of both models are marked in this figure.

The deformation contour spacing of the parametric model in the L1 positive direction gradually decreases from the maximum deformation zone, while that in the L2 positive direction decreases initially and then increases. There is the same changing law of the deformation contour in the simplified model. Here, 30% of the maximum deformation is defined as the large deformation zone, which is marked as a red solid line in the simplified model and a blue dashed line in the parametric model. It is obvious that the total areas regarding both large deformations are similar. With the decrease of the deformation, the contours of both models are closer. This indicates that the simplified model can also fit a relatively accurate deformation-changing law outside the large deformation zones.

Obvious differences between both models are the maximum deformation positions and large deformation zones. The maximum deformation position in the simplified model is almost in the center of the roof surface. With the application of the parametric model, this position moves to the positive direction of L1 and L2. However, the deviation of two maximum deformation positions in the L2 direction is not as obvious as that in the L1 direction. With the combination of the centroids in

Figure 13, the relative position of two centroids is opposite to the relative position of the maximum deformation points. The reason focuses on the effect of the centroids’ position and values of moments of inertia in different directions. With the movement of the centroid to the left and lower sides, here from the simplified model to the parametric model, the anti-deformation ability in the left and lower zones improves in the parametric model. Consequently, the large deformation zone of the parametric model moves to the right and upper direction when compared to the simplified model. The large deformation position changes more obviously in the L1 direction than in the L2 direction. This is due to the geometric characteristics of this object. From the relative perspective in

Figure 13, the left and right sides are, respectively, the south and north surfaces of the building. Their architecture and locations are more symmetrical. However, due to the existence of the obvious concave part in the upper zone, this feature structure can better support the loading force of the object. Therefore, the large deformation zone does not move upwards obviously.

#### 4.4. Stress Analysis Based on Static Process

Equivalent stress is of great significance to monitor the strength of the damage possibility in the loading progress, as it is shown in

Figure 16.

The color difference in the deformed zones of the surfaces is obvious. This indicates the stress change. There are two concave strip shapes in the red enlarged area of

Figure 16, while the equivalent stress contour of this zone is smooth in the simplified model of

Figure 16b. The equivalent stress of the upper concave peak part is 125.24 MPa, while the equivalent stress of the concave valley part is 123.93 MPa. The equivalent stress of the lower concave peak part is 131.99 MPa, while the equivalent stress of the concave valley part is 130.49 MPa. Three line examples are extracted in the same concave change part. The length is about 0.2m which is close to the width of the concave region. The equivalent stress along the line length is listed in this figure. It indicates that the deformed concave shape affects the stress results. By contrast, this detailed feature cannot be found on the flat surface of the simplified model. It indicates that the deformed surfaces have a direct effect on the surface stress behaviors in the loading progress. There is a great difference regarding the maximum equivalent stress between the parametric model and simplified model, which is enlarged as the black dashed frame zones in

Figure 16. Two black enlarged zones in the simplified model show that the maximum equivalent stress on the roof surface is close to the maximum value of the global model. However, the maximum stress of the parametric roof surface is far less than the global maximum stress value. The damage behavior in the engineering application will be predicted inaccurately if the parametric model is ignored.

The contour shape regarding the large stress zone of both models is generally similar. There is a similar development law of the equivalent stress contour if the specific value is not considered seriously. Consequently, similar models contain similar stress development laws in the same conditions. Hence, the common simplified model can be reasonably adopted in the computation of structural problems when the research focus is on the overall laws because the simplified model is more efficient.

#### 4.5. Vibration Analysis

Modal analysis is the basis of the dynamic structural computation. With the help of the modal analysis, it is helpful to recognize different responses of the structural problems to different dynamic loadings. The vibration analysis involves a wide range of knowledge. However, this research focuses on the analysis of two reconstructed geometric models. Therefore, only a contrast dynamic analysis between two models is carried out based on the frequency and the vibration deformation.

The relative modal data in

Table 5 are beneficial for comparing the vibration characteristics of the structural problems [

71]. Error 1 calculates the relative deviation regarding the maximum vibration deformation between the parametric and simplified models. Error 2 calculates the relative deviation regarding the modal frequency of two models. Two errors are obvious, which implies the inadequacies of the common simplified model. By contrast, the maximum deformations of the parametric model in the 1st, 2nd, and 3rd modal conditions are far less than those of the simplified model, while the frequencies in the parametric model are higher than in the simplified model. This indicates that the parametric model is more capable in the dynamic computation of structural problems.

## 5. Conclusions

This manuscript offers a generic methodology which focuses on the FEA based on the parametric model by approximating 3D actual feature data. 3D actual feature data can be acquired from many efficient measurement methods, e.g., TLS, digital photogrammetry, and radar technology. The methods in approximating parametric surface models accurately are diverse, e.g., the T-spline method, the B-spline method, and NURBS approximation.

It was applied to detect the deformed surface information by TLS which is an accurate and reliable measurement technology in this research. The B-spline method was applied to approximate the measured point clouds data and generate the parametric 3D model of structural problems. The parametric model can be applied in both CAD modelling and CAE analysis innovatively. The model quality and some deviations were discussed. The static and dynamic computations were carried out to imply the advantages and disadvantages of both models regarding the responses of different loadings. The main conclusions are as follows.

1. The numerical parametric model of structural problems satisfies the continuity characteristic in constructing the 3D model with the advantage of the parametric description method B-splines.

2. The parametric model can reserve deformed features of the structures accurately. When the data measured is compared, both the maximum and standard deviations of the parametric model are far less than the simplified model. The mass volume of the simplified model is smaller than that of the parametric model due to the lack of the description of deformed walls. This is one of the main reasons why there are obvious deformations between the parametric and simplified models in the static and dynamic computations.

3. In the static structural analysis, the simplified model is acceptable and it is more convenient and efficient to analyze and predict the overall development law regarding the deformation and stress outside the large deformation zones. The simplified model has obvious shortcomings and inaccuracies in large deformation zones compared to the parametric model in the static structural computation.

4. The FEA computation based on the parametric model is more reliable which benefits from the parametric description of the actual object, e.g., B-Spline method.

5. The parametric model contains fewer errors in the dynamic computation. It indicates that the parametric model is more reasonable and acceptable in the dynamic computation and analysis, especially in the large deformation and stress zones.

6. There is an obvious effect of predeformed parametric surfaces on the equivalent stress of the composite structures in the loading progress. Therefore, the parametric model is more accurate in predicting the future behavior by analyzing the equivalent stress of composite structures.

Attention should be paid to the fact that the FEA model simplification can bring some efficiency to researchers to a certain extent, e.g., when the research purpose is only to predict some simple regularity problems or development trends. However, numerical features, e.g., the deformation, the stress, and the strain, should be focused on significantly in most cases when it is in the accurate design, optimization, and prediction stages. Therefore, the FEA model simplification is not desirable in this case and the accurate parametric model is significantly desired.

In summary, the FEA based on the parametric model is instructive for understanding the structural progress and predicting the damage behaviors in this paper. The novel parametric model method is efficient and beneficial to improve the reliability and accuracy of the FE computation regarding the composite structural problems. Moreover, it is of great significance to apply the FEA parametric model to health to monitor the future damage behavior of the composite structures in the engineering application.