**3. Results**

### *3.1. Covergence Analysis*

To show the accuracy of the methodology, problem (1) is solved using the proposed Immersed Boundary Robin-based (IBR) model, on the idealized geometry domain of Figure 6B, for an internal pressure of *p* = 10−<sup>2</sup> MPa. Consequently, the analytical solution of the problem is calculated for the infinite solid with a cylindrical cavity at *rint* (Figure 6A). Recall that, to make equivalent both the analytical results with the Robin-based approach,

the *ballast* coefficient *α* of Equation( 19) is used. The different model parameters used for this study are reported in Table 2.

**Table 2.** Model parameters: Young's modulus *E*, Poisson ratio *ν* and the *ballast* coefficient *α*.


Figure 7A represents the relative error of the displacements at an arbitrary point with radius *r* ∈ {*rint*,*rext*} (compared to the analytical solution obtained using Equation (12)) of both the proposed Immersed Boundary Robin-based approach (in black) and the classical FE approach in red (which is calculated for completeness in the analysis).

**Figure 7.** Relative error of (**A**) displacements magnitude and (**B**) TDE obtained for the IBR (black) and classical FE (red) formulations.

In addition, recalling Equations (20) and (21), the relative error (with respect to the analytical value) of the total deformation energy (TDE) is also calculated for both the IBR and the classical FE models and represented in Figure 7B in black and red, respectively. It is worth noting the convergence behavior by increasing the number of active nodes, which means refining the conformal mesh for the classical FE approach and the unfitted background mesh for the IBR solution; the solution improves faster for a smaller number of active nodes using our IBR methodology until a stabilization plateau, where the relative error of TDE stagnates at a value close to 10−4. This is likely due to truncation errors affecting quantities computed in the unfitted procedure, mainly integrals in elements divided by boundaries. This level of accuracy is perfectly acceptable in this type of model.

#### *3.2. Elastic Bed Coeficient α: Sensitivity Analysis*

As mentioned in Section 2.1, the elastic bed coefficient *α* represents an elastic bed boundary condition that simulates the interaction of the body with its surroundings. Therefore, it is also possible to tune *α* so that it only avoids rigid body motions (and not influencing nodal displacements and stresses). For this study, Figure 8A shows a realistic arterial section where the proposed *α* analysis is applied under an internal pressure *p* = 2.6267 × 10−<sup>2</sup> MPa. Homogeneous linear elastic material properties were used, being the material parameters reported in Table 3.


**Table 3.** Material parameters of the piece-wise homogeneous domain Ω = 24 *k*=1 Ω*k* (i.e., normal vessel wall, loose matrix, calcification, and lipid core), *E* (Young's modulus) and *ν* (Poisson ratio).

According to this, the graph in Figure 8B shows that by decreasing the elastic bed coefficient *α*, the average displacements at the external boundary ( **<sup>u</sup>**|<sup>Γ</sup>*R* ) becomes unaffected by the surroundings while eliminating rigid body motion. That is to say, the left part of the graph corresponds to the limit value of a floating object with no surrounding stiffness: the small values of *α* do only suppress rigid-body modes. The limit case for large values of *α* has zero displacements in the external boundary, as it is reflected in the plot.

**Figure 8.** (**A**) Realistic arterial (coronary) section domain Ω = 24 *k*=1 Ω*k*. (**B**) Elastic bed coefficient against average displacement on Γ*R* for the coronary section under an internal pressure *p* = 2.6267 × 10−<sup>2</sup> MPa.

#### *3.3. Characteristic Length h of the Background Mesh* T*h*(Ω*<sup>B</sup>*)*: Sensitivity Analysis*

A relevant parameter in the IBR methodology is the characteristic length *h* of the background mesh. It is strictly related to the number of active nodes (*nact*) by inverse proportionality, meaning the smaller the *h* value, the greater the *nact* value. It is possible to verify how the IBR solution, in terms of displacements and TDE, converges to those obtained with the classical FE method using a very fine mesh. For this, six background meshes <sup>T</sup>*h*(*i*)(Ω<sup>B</sup>) *i* = 1, . . . , 6 are used with a decreasing value of *h* as *i* increases.

Figure 9 shows the results of such analysis for the section depicted in Figure 8. Figure 9A reports the error for both displacements and TDE associated with the six background meshes with respect to the conformal mesh solution, where it is shown the clear error decreases by increasing the number of active nodes. As an example, Figure 9B shows a plot of the local displacements difference error associated with the finest background mesh developed for this analysis. Table 4 shows the maximum differences in TDE and the displacement magnitude for the different background meshes considered in the analysis.

**Figure 9.** (**A**) Convergence profile for the maximum difference for TDE and displacement magnitude. (**B**) A plot of the local difference in displacement magnitude for a subset of the nodes in the conformal mesh. In particular, B refers to mesh 6 from Table 4.

**Table 4.** Results of the sensitivity analysis about the characteristic length *h* for TDE and displacement magnitude. From mesh 5, there are more active nodes (*nact*) in the background than nodes in the conformal mesh (*nact* conformal mesh = 64, 552).


### *3.4. Realistic Immersed Boundary Robin-Based approach*

Figure 10 shows a comparison between the IBR methods and the classical FE approach for a realistic coronary section, described in Figure 10A, subjected to an internal pressure of *p* = 10−<sup>2</sup> MPa. The model parameters used for this study are shown in Table 5, being the material properties reported in Table 3.

**Figure 10.** Realistic arterial (coronary) section (**A**) domain Ω = 2<sup>4</sup>*k*=<sup>1</sup> Ω*k* displacements distribution (**B**) using IBR and (**C**) classical framework. Panel (**D**) depicts relative local error for displacements.


**Table 5.** Background mesh T*h*(Ω<sup>B</sup>) and conformal mesh T*h*(Ω) parameters.

Figure 10B shows the displacement field obtained with the proposed Immersed Boundary method (obtaining a total deformation energy TDE = 6.8226 × <sup>10</sup>−3), while Figure 10 C corresponds to the solution obtained with the classical FE method (TDE = 6.8248 × <sup>10</sup>−3). Differences in the total deformation energy are found to be less than 0.05%, with a maximum difference in the displacement magnitude of less than 5% (Figure 10D), being an average error of less than 0.5% in the section. Figure 11A,B show the distribution of the Von Misses stress obtained for both methodologies, IBR method, and the classical FE method, respectively, showing a highly similar stress distribution.

**Figure 11.** Realistic arterial (coronary) section domain Ω = 24 *k*=1 Ω*k* Von Misses stress distribution using ( **A**) IBR methods and (**B**) classical framework.
