2.3.1. Boundary Value Problem Analysis Using only the PDM
PCD consist of arbitrary points located on the surface of a specific object. To analyze a BVP, it is necessary to impose essential boundary conditions at the boundary of the PDM model. Typically, numerical and structural analyses are performed by directly applying displacement boundary conditions and loads. However, in this study, the deformed shape of the structure is considered to contain information on the kinematic variables of the interior nodes. Therefore, we validate a methodology that applies the deformed shape, tracked by LiDAR, as a boundary condition to the PDM model. In this section, before verifying the hybrid simulation technique, the accuracy of converting the deformation shape of the structure into the PDM essential boundary conditions is validated using a 2D elastic beam problem for which an analytical solution is known, as shown in
Figure 2. Equations (15) and (16) are analytical solutions for a linearly elastic cantilever beam subjected to an end load, as illustrated in
Figure 2. More details on this problem can be found in Timoshenko and Goodier [
29].
The closed-form solutions for stresses are also given by
where
is the moment of inertia for a rectangular beam cross-section.
and plane stress are assumed with Young’s modulus
, and Poisson’s ratio
.
Before validating the methodology of extracting essential boundary conditions from structural deformation shapes, we conducted an accuracy analysis of the numerical solution obtained using the PDM for the essential BVP. In fact, the deformation shapes or essential boundary conditions for the four edges of the cantilever beam can be obtained from Equations (15) and (16). These boundary conditions assigned essential boundary values to the PDM boundary nodes. The analysis was performed to evaluate the error, and the relative sup-norm error in displacement was calculated as follows:
where
error refers to the sup-norm error, representing the most significant value among each node’s calculated errors. The subscript
ex denotes the exact solution, and
PDM refers to the results calculated using the PDM.
Figure 3a,b illustrates the relative errors in the
x and
y-directional displacements of the PDM nodes plotted as surface plots, respectively.
Meanwhile, the PDM analysis results can also be evaluated using an
error norm, as shown in Equation (19).
where
Ω refers to the problem domain,
represents the displacement vector calculated from the PDM analysis results, and
refers to the displacement vector following the closed-form solution in Equations (15) and (16). The convergence study depicted in
Figure 4 demonstrates the rates in the
error norm as the node spacing of the model, which is distributed at regular intervals, decreases. As anticipated, it is observed that the PDM BVP analysis algorithm performs with high accuracy as the number of analysis nodes increases. This indicates that if the essential boundary conditions for the structural boundaries are correctly provided, the stress inside the structure can be precisely calculated using the PDM.
2.3.2. Polynomial Regression and Essential Boundary Condition Capturing for Boundary Value Problem Analysis Using PCD
This section presents a method for constructing a function formula to assign essential boundary values to the PDM boundary nodes. The essential boundary values are obtained from the deformation shape of the structural boundary extracted from PCD. A technique that utilizes the eigenvalues of k-nearest neighborhood points [
30] was applied to estimate boundary edge points from PCD. More detailed information about this technique is provided in
Section 3, which also includes the process of handling PCD obtained from a 3-point bending test of an actual rubber beam.
A method for assigning displacement boundaries to boundary nodes of a PDM model is presented in
Figure 5. The figure depicts the deformed shape of a two-dimensional cantilever beam from
Figure 2, where edge points were calculated by adding
and
values given by Equations (15) and (16) to the undeformed boundary coordinates. The displayed deformation shape was obtained by multiplying the
and
values by 50 to exaggerate the deformed shape. This approach for capturing essential boundary conditions can be applied to the hybrid simulation technique for the actual structures using the PDM-based BVP analysis.
Owing to the nature of LiDAR, the PCD acquired from the same target surface at different times will inevitably have a different number of points. Even if the number of points is the same, these values are 3D coordinates of randomly reflected laser points, so comparing the relationship between points before and after deformation with the initial PCD is impossible. Therefore, to determine deformation history or displacement at a specific location, fixed targets that can be verified at specific locations [
31] or specific point changes are often utilized as measurement features [
7]. However, this paper proposes a new technique that can track the shape changes of the structural boundary without using fixed targets or other features on the target structure. This process is schematically represented in
Figure 6, with Step 1 being the approximation of the edge deformation shape using polynomial regression.
Below is the general form of the one-dimensional polynomial regression equation applied to track the shape changes of the deformed structural boundary.
where
x is the independent variable,
y is the dependent variable,
n is the highest degree of the regression equation,
is the coefficient vector of the polynomial regression equation, and
is the polynomial vector. We select only one independent variable for a one-dimensional polynomial to describe the shape of the deformed boundary’s boundary while avoiding complex function expressions. For the cantilever problem, an accurate function expression representing the boundary deformation shape would depend on both
x and
y coordinates, as observed in Equations (15) and (16). However, choosing a single independent variable with a dominant influence is more efficient. Therefore, this section constructs the polynomial regression equation concerning the more dominant coordinate axis.
The coefficient of determination (
R2) of the regression equation [
32] may vary depending on whether the
x or
y coordinate is selected as the independent variable in the Cartesian coordinate system of the edge points. For instance, in
Figure 5, when a straight line such as the right side of the beam (line 3) remains almost straight after deformation, the
R2 value does not significantly differ based on the choice of independent and dependent variables. However, to represent deformation shapes such as the left side of the beam (line 2), selecting the
y coordinate as the independent variable yields a higher
R2 than selecting the
x coordinate, necessitating a third-degree or higher regression equation. This is because the
y coordinates’ distribution of edge points is constant from −6 to 6, while the
x coordinates’ distribution varies between positive and negative values. In this case, the analytic solution for displacement
ux and
uy is expressed as a combination of
x and
y up to the second-degree terms (including cross terms). However, since the function that can describe the deformation shape of an actual structure from PCD is unknown, it is necessary to idealize the deformation through appropriate assumptions. Thus, this section defines the cantilever beam deformation as a one-dimensional polynomial with a single independent variable to simplify the equation. For convenience, the independent variable of the polynomial regression equation is set to the coordinates along the edge-axis parallel to the edge direction.
Edge points extracted from PCD obtained through LiDAR scans are complex and unevenly distributed, and most contain outliers. Therefore, it is crucial to avoid overfitting when choosing a regression equation with a high R2 and to balance precision and efficiency when assuming and calculating an appropriate polynomial. Once the regression equation is calculated, it contains information about the coordinates of the nodes after deformation, and by knowing the coordinates of the nodes before deformation, it is straightforward to calculate the displacement boundary condition.
Among the PDM boundary nodes, corner points of a structure can be determined from the intersection points of regression equations without any additional assumptions. For example, in the case of a rectangle such as that in
Figure 5, the
x coordinates of the four corner points can be obtained from the intersection points of the regression equations with the same independent variables for the four edges. The
y coordinates can then be obtained by substituting these values into the regression equations, determining the two-dimensional coordinates of the corner points after deformation. These values are then used as initial values for determining the essential boundary values of the PDM boundary nodes between the corner points in
Figure 6, Step 3.
Table 1 summarizes the polynomial regression results for the edges in
Figure 5. The RMSE (Root Mean Square Error) column represents the average prediction results. As mentioned earlier, both
x and
y coordinates of the exact solution were used in the regression process, most
R2 values being close to 1. However, the lower
R2 value for the
x independent variable regression equation for line 2 compared to others is due to the trend mentioned above for line 2. In such cases,
R2 can act as an error indicator when determining corner point displacements derived from the intersection finding process of other regression equations.
In this study, the independent variable of the regression equation was set as the edge-axis coordinates. Therefore, if the edge-axis displacement of the PDM boundary node can be determined, the edge-axis coordinates after the deformation of the PDM boundary node can also be determined. These coordinates are used as the independent variable values in the regression equation, and the coordinates in the direction perpendicular to the edge-axis are the dependent variable values calculated through the regression equation. In other words, by using the estimated edge-axis displacement for the PDM boundary nodes, the coordinates of all the PDM boundary nodes after deformation can be determined; displacement of the PDM boundary nodes can be derived by subtracting the coordinates of the PDM boundary nodes after deformation from the coordinates. When the edge-axis is the
x-axis, the independent variable coordinates
of the regression equation can be expressed as follows:
where the superscript denotes the number of the edge nodes, with the starting point as zero and the endpoint as
n. The independent variables indicating the coordinates after deformation along the edge-axis at the
ith edge node are denoted by
, where the edge-axis coordinates of the PDM boundary node are
and
represents the edge-axis displacement determined through the displacement allocation function described later. In Step 3 of
Figure 6, the displacement along the PDM boundary nodes is calculated through the edge-axis displacements
and
determined in Step 2. Although the relationship between the edge-axis displacement and
is unknown, it is assumed to have a polynomial function relationship. This assumes a polynomial relationship between the pre-deformation and post-deformation; if this relationship is expected to be nonlinear, a higher-dimensional displacement mapping and tracking method would be implemented. This study assumes this variable is a quadratic function approximated as a cumulative sum of strain.
Figure 7 illustrates the method of calculating the edge-axis displacement, assuming linear strain at an arbitrary position for the end values,
and
with zero strain at
. To simplify computation, it is assumed that the strain is constant within a section divided into equal sizes, and the value at the center of the section is used.
The edge-axis displacement can be represented as a quadratic function of
as follows:
where
a,
b, and
c are coefficients of the quadratic polynomial
,
is the displacement at the starting point of the given interval,
. For example, if the edge-axis is coaxial with the
x-axis and the spacing of edge-axis nodes in the PDM model is uniform, the nodal displacement
of the
kth node can be given as the cumulative sum of strains, which is given as follows:
where
is the sum of strain between the
-th and
th intervals, equivalent to the area within the unit interval of the strain graph in
Figure 7. If the PDM analysis nodes have uniform spacing, the difference between
and
is equal to
, which is the increment in the strain. In the last interval,
is equal to the triangular area where the strain graph intersects the edge axis, and it is equal to
. Therefore, accumulative strain over the entire interval is given by
where
is the edge-axis displacement over the entire interval, which is the difference between the end values
and
.
Figure 8 illustrates the displacement and strain for the assumption of edge-axis displacement as a quadratic function (
Figure 8a,c,e) and a linear function (
Figure 8b,d,f). In the case of assuming a linear function including a bi-linear function, the magnitude of the edge-axis nodal displacement can be represented as a linear equation, as shown in
Figure 8b,d,e. The resulting function depends on the regression independent variable (Equation (21)). The displacement in the direction perpendicular to the edge-axis of the analysis point is related to the degree of the regression equation. This approach is suitable for modeling cases where strain occurs uniformly, such as uniaxial tension, as it might assume constant strain.
The deformation pattern in
Figure 8a is similar to that of line 1 or 4 in
Figure 5; the additional condition that the axial strain at the free end is zero can be used to complete the edge-axis nodal displacement equation. In this case,
in Equation (23) is obtained as follows:
where
is the value for
. Rearranging Equation (24) results in
where
=
. Since the above formula can be used even when
is negative by substituting it into Equation (25), it is also valid for modeling line 4.
In
Figure 8c, the scenario is similar to the case of line 2 in
Figure 5, where the edge-axis displacement is the same at both ends and the total displacement of the entire interval,
, is assumed to be zero. Without additional conditions, it is impossible to determine ∆g. However, since line 2 is constrained at the center point with zero axial displacement, the quadratic polynomial for deformation modeling becomes symmetrical based on this point, and the strain at the center point is zero. By using this condition, the strain graph shown in
Figure 8c can be derived, and the value of
in Equation (23) can be expressed as
where
relates to the value
according to Equation (28).
where displacement is the sum of the deformation amounts of two sections concerning a fixed central point. Hence, the total displacement
for the entire section of line 2 is
. In this case,
=
. As observed in Equation (23), when it is the
nth node,
becomes
. Therefore, using this equation, an edge-axis nodal displacement equation with a quadratic form can satisfy the condition that the displacement for the entire section of line 2 is zero.
Figure 8e illustrates a case where the edge-axis displacements at the starting and ending points occur in different directions. The nodal displacement curve in
Figure 8e shows that since the directions of edge-axis displacements at the starting and ending points are different, the displacement amount should be zero at some point within the section. Therefore, unlike
Figure 8a,c, it is impossible to represent the edge-axis displacement equation for the entire section with a single quadratic curve. However, using the condition that the point where strain becomes maximum and the point where
becomes zero are the same, an edge-axis nodal displacement equation with a quadratic form for two sections centered on the central point can be constructed according to Equation (24) as follows:
where
relates to the value for
with
=
. At this time, the sectional
is given as follows:
In this case, the slope of the strain graph changes based on the maximum strain point. Hence, Equation (23) can be arranged for each section as follows:
The deformation amount for the PDM boundary nodes is determined by constructing the edge-axis displacement equation, which can be directly used as essential boundary conditions in the PDM analysis. Therefore, the advantage of using the method proposed in this study is that it can determine the essential boundary condition values required for BVP analysis, even when there is no target on the specimen surface or when the feature characteristics distinguished by PCD are ambiguous. The following section presents the results verifying the developed analysis technique by simulating an actual three-point bending test.
2.3.3. Verification of the Developed Analysis Technique through an Elastic Beam Problem
This section presents the results of applying the essential boundary condition values to the PDM model. The boundary condition values were determined based on the pre- and post-deformation boundary shapes of the structures in
Figure 5 and the procedure outlined in
Figure 6 or the previous section. Stress was calculated by using the PDM-based BVP analysis. We applied the regression results in
Table 1 to assign the cantilever beam’s essential boundary condition (line 1–4). We investigated the convergence of the two displacement capturing methods (linear strain model, uniform strain model) and the number of nodes of the PDM model. For line 3, which can be sufficiently approximated using a linear equation, as seen in
Table 1, we applied the linear regression method and distributed the displacements uniformly.
Figure 9b,c shows the surface plots of the
x-directional displacement calculated by the PDM analysis for different displacement capturing methods. The results that were calculated assuming linear strain were similar to the exact solution, while those calculated using the uniform strain model displayed some differences. These differences arose because the edge-axis displacements determined by the two methods differed. However, as observed in
Figure 9e,f, the trends of the
y-directional displacement distributions were similar regardless of the degree of the regression function. However, variations in the values of the dependent variables due to differences in the independent variables resulted in variations in the values of the kinematic variables.
Figure 10 and
Figure 11 compare the stress values calculated using the two displacement capturing methods with the analytic solution in Equation (17). In
Figure 10, the application of the linear strain model for line 1 and line 4 resulted in some differences from the analytic solution, as indicated by the red dotted circles, but the differences were insignificant.
Figure 10b,c presents the calculation results for
, which were similar to the trends observed in the strain models of
Figure 8a,b. The linear strain model provided displacement for the PDM node by accumulating a strain per section, resulting in stress resembling the strain distribution shown in
Figure 8a. On the other hand, the uniform strain model considered a constant edge-axis strain, resulting in the
calculation results shown in
Figure 10c. In other words, the differences in the displacement capturing method associated with the regression method were linked to the differences in the stress calculation results.
Figure 11b,c presents the calculated von Mises stress results. The results obtained using the linear strain model in
Figure 11b well matched the analytic solution, whereas the results obtained using the uniform strain model showed a considerably smaller von Mises stress value at the beam corner than the analytic solution. Additionally, the error occurrence pattern in the internal area was quite different, as shown in
Figure 11c.
Figure 12 presents the surface plots of the relative sup-norm error of the displacement (Equation (18)) for the two displacement capturing methods. The results of applying the linear strain model in
Figure 9a,c were similar to those in
Figure 3, which was the traditional BVP analyzed using the PDM. However, the more significant error in the corner part of
Figure 9a compared with
Figure 3a was due to difficulty determining the corner points where the regression equations intersected. The errors primarily occurred around Corner Point 1 and Corner Point 3 in
Figure 5, near the support of the cantilever beam, where the
x-directional displacement was fixed. However, as the polynomial regression equations accurately approximated the deformed shape, the absolute value of the resulting error was not significant, and its impact was limited to the nodes near Corner Point 1 and Corner Point 3, as shown in
Figure 12a. In contrast, for the uniform strain model presented in
Figure 9b,d, the error magnitude at the corner points was smaller compared to other areas, but relatively much larger errors were observed in the middle part of the edge between corner points.
Figure 13 presents the results of a convergence study conducted using the error above analysis scheme. The plot shows that both methods of setting the node intervals exhibited convergence, as the error decreased when the number of nodes increased. Furthermore, the absolute error magnitude was significantly smaller for the linear strain model than for the uniform strain model. Therefore, when a straight edge becomes curved after deformation, applying the linear strain model can improve the accuracy of calculating the kinematic variables of the entire domain when solving BVP using the PDM.