4. Soft object design of TPS

#### 4.1 Rationale

The modeling procedure for the TPS is defined starting from the definition of a set of soft objects which are represented on the topological map associated with the current morphology of the object, as shown in Figure 3. Consequently, the supports of the sticks are adjusted according to the normalized dimensions relative to this map. The topological map is emulated introducing a two-dimensional grid (from

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

#### Figure 3. Morphological (left) vs. topological map (right).

now, denoted as B-grid) having the same topology tree than the vehicle open grid (number of points, panels, and connectivity) but unit size. A geometric mapping between the B-grid and the vehicle grid is established, and elements of B-grid are univocally mapped onto corresponding elements of the vehicle surface (see Figure 3). Therefore, each centroid of panels which belong to topological map has the same neighboring points either on the topological or morphological map. Several stick primitives are emulated on B-grid placing a number of n equally spaced isotropic blobs, with radius r and length l, respectively, in a normalized unit. Stick emulation is performed by overlapping n blobs using the special formulation reported in [17] that ensures a convergent envelope of the finite support and a limited value of the blob strength. An exemplificative spatial distribution of sticks on the B-grid is shown in Figure 3.

Position and orientation of each stick are determined by assigning coordinates of centers Ci and precession angles θi, respectively, with respect to a Cartesian frame of reference Oxz oriented as in Figure 3. Therefore, a generic distribution of sticks created on vehicle grid is equally mapped on the vehicle surface whatever is the morphological map considered. In the present case, gray-colored regions (1) denote points of the B-grid mapped on the windward side of RLV shape (see Figure 3), while white regions (2) relate to leeward regions of the vehicle. Regions of vehicle surface mainly subjected to heating peaks during the reentry maneuver are (i) nose, (ii) leading edge, and (iii) tail. The global potential field generated by the sticks onto the B-grid is adjusted in a suitable dimensional scale and subsequently mapped on the mesh panels of the vehicle surface grid to obtain an easy and powerful control of the thickness distribution. The proposed methodology is able to create virtually arbitrary TPS distributions and can be easily tuned up to locally increase the thickness where critical heat loads are expected and dropping out elsewhere. A similar, slightly modified procedure is also applied to create an arbitrary binary map

more regular, but the strength of the field function diverges. The above drawback is overcome modifying the definition of potential field given by Eq. (2) with the

Equation (9) where F0(P) = 0 expresses the global potential field Fj(P) irradiated by a set of j blobs at a generic point P of space placed at a distance d from the key points, as the max between the previous j � 1 potentials accounted by the assembly layer Fj � 1(P) and the current potential Gj over the plane disk of radius r:

Gjð Þ¼ <sup>P</sup> f Pð Þ <sup>d</sup><<sup>r</sup>

0 otherwise

Figure 2a, b shows the support and the strength field of a two-dimensional stick primitive obtained with nblob = 6 and 20, respectively, computed with Eq. (8). By increasing the number of blob on a stick, the strength of F is still bounded to a maximum unit value. Figure 2c, d shows the same behavior for a tapered primitive having a linear variation of the blob radius along the axis of stick. Therefore, a seamlessly blending of blobs, with a bounded strength, is obtained adopting Eq. (9). The procedure proposed here relies on a similar idea to the one developed in [17] to generate self-stiffened structural panels. Specifically, rather than modeling an object tracking an iso-contour of its potential field, the full integral field generated by a set of blobs spatially arranged on a two-dimensional grid generates a smoothly

A generic shape of an RLV is represented by a grid formed by a quadrangular and/or by either degenerated triangular panel grid. Grid points are obtained using a proprietary procedure that authors fully detailed in [20, 21]. Without going into details of the shape model, we remark that the mesh arrangement over the RLV surface is obtained with no NURBS support surface: a three-dimensional parametric wireframe is created using cubic rational B-splines [22] and used to reconstruct computational surface grid. The control parameter allows a wide range of shape variations to handle different design objectives (thermal or dynamical) for a reentry mission. Grid topology is equivalent to a spherical surface with no singularities (open poles) and allows a mapping of the points in UV coordinates over an equivalent cylindrical surface. The above considerations ensure a topologically invariant

The modeling procedure for the TPS is defined starting from the definition of a set of soft objects which are represented on the topological map associated with the current morphology of the object, as shown in Figure 3. Consequently, the supports of the sticks are adjusted according to the normalized dimensions relative to this map. The topological map is emulated introducing a two-dimensional grid (from

Fj�1ð Þ <sup>P</sup> ; Gjð Þ <sup>P</sup> <sup>j</sup> <sup>¼</sup> <sup>1</sup>, <sup>⋯</sup>, nblobs (9)

(10)

Fjð Þ¼ P max ∀P

Hypersonic Vehicles - Past, Present and Future Developments

relation:

varying field.

shape.

20

4.1 Rationale

3. RLV shape modeling

4. Soft object design of TPS

Figure 4. Arbitrary stick distribution created over the topological map.

distribution of different TPS materials that may be operated independently of the thickness distribution. Figure 4 shows an arbitrary distribution of stick primitives (not suitable for application purposes) created over the topological map.

The resulting potential field created by the superposition of sticks modulates y-coordinate of grid points as shown in Figure 4.
