5. Parametric model of thermal protection system

#### 5.1 Thickness modeling

As demonstrative example, a parametric representation of TPS is obtained using a limited set of sticks primitive (nstick = 5), oriented as shown in Figure 5. Skin sticks characterized by a large radius and limited strength are spread over the skin surface in longitudinal direction in order to provide a thickness graded baseline. A constant minimum thickness is superposed in all remaining points of B-grid, ensuring a nonzero value in any point of the grid. Furthermore, additional parametric sticks, specifically positioned and oriented to affect thickness in critical regions as nose, leading edge, and trailing edge, complete the support for TPS and create a rational distribution of insulating material suitable with a reentry mission. Parametric position of sticks and axis of orientation are defined by assigning centroid coordinates xc,zc and angle θth, measured with respect to the system of reference reported in Figure 5. Length (l) and strength (th) are expressed with the parametric relations

$$\begin{cases} \mathcal{X}\_{c,\{q=1,2,3,4,5\}} = \{0.0, 0.0, 0.0, 1.0, 1.0\} \\\\ \mathcal{z}\_{c,\{q=1,\dots,5\}} = d\_{q\_{\min}} + st\_{q} \bullet \left(d\_{q\_{\max}} - d\_{q\_{\min}}\right) \\\\ \mathcal{l}\_{\{q=1,\dots,5\}} = \mathcal{l}t\_{q} \bullet d\_{q\_{\max}} \\\\ \mathcal{th}\_{1} = t h'\_{\min} + p t\_{1} \bullet \left(t h'\_{\max} - t h'\_{\min}\right) \\\\ \mathcal{th}\_{\{q=2,\dots,5\}} = t h''\_{\min} + p t\_{q} \bullet \left(t h''\_{\max} - t h''\_{\min}\right) \end{cases} \tag{11}$$

5.2 Material modeling

Figure 5.

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A similar but completely independent stick-based parameterization has been also defined to model a dynamic distribution map of different insulating materials, denoted here generically as material 1 and material 0 represented with red and blue colors, respectively. We assume that material 1 outperforms material 0. Therefore, material 1 is adopted on the nose, leading edge, and trailing edge, respectively. Differently than sticks used for thickness distribution, this additional set of primitives returns just binary values used to define specific materials. In this case the field function mth (see relation (12)) assumes a constant value equal to one inside the finite support of a stick and zero elsewhere. The parametric equations which

Arbitrary stick distribution with a longitudinal gradient onto B-grid adopted for TPS modeling.

Parametric Integral Soft Objects-based Procedure for Thermal Protection System Modeling…

DOI: http://dx.doi.org/10.5772/intechopen.85603

mxc, f g¼ <sup>q</sup>¼1;2;3;4;<sup>5</sup> f g <sup>0</sup>:0;0:0;0:0;1:0;1:<sup>0</sup>

mlf g <sup>q</sup>¼1;…<sup>5</sup> ¼ mltq ∙ dqmax

mthf g <sup>q</sup>¼1;…;<sup>5</sup> ¼ 1

mzc,f g <sup>q</sup>¼1;…;<sup>5</sup> ¼ dqmin þ mtq ∙ dqmax � dqmin

� �

(12)

Skin (q = 1, 2) and nose sticks (q = 3) have a tapered support obtained imposing a linear variation of point source blob radius. Conversely, a constant radius is adopted for the leading edge (q = 4) and trailing edge (q = 5) sticks.

Parametric Integral Soft Objects-based Procedure for Thermal Protection System Modeling… DOI: http://dx.doi.org/10.5772/intechopen.85603

Figure 5. Arbitrary stick distribution with a longitudinal gradient onto B-grid adopted for TPS modeling.
