2. Soft objects definition

[3, 4]. In order to fulfill all those requirements, the duration of reentry flight increases and consequently the integrated heat load absorbed by the structure [3]. The above consideration incidentally demands a trade-off among several nonlinear conflicting design objectives, also satisfying a number of constraint functions. As an example, the design of the TPS of an RLV performing a suborbital lifting reentry requires a mandatory compromise between the maximum allowed peak heating and the integrated heat load. This requirement may conflict with the adoption of a fully reusable TPS, either limiting the choice of material category or penalizing the total mass. In preliminary design practice, thousands of design configurations are typically evaluated by an optimization algorithm to find the best fit [5–11]. Therefore, a preliminary appraisal of vehicle performances is commonly performed using high-efficiency, low-order fidelity methods that give a support to a multidisciplinary analysis performed with a computational effort which fit the typical timeline of the conceptual design phase [11]. In current studies, TPS sizing is performed using several simplified assumptions, carrying out a one-dimensional heat conduction analysis with panel thickness modeled using stackups of different

Hypersonic Vehicles - Past, Present and Future Developments

The aerothermal environment is a basic design criterion for either TPS sizing or

choice of materials [13, 14]. Several works dealing with TPS sizing have been published in literature. Lobbia [8] determined the sizing of a TPS in the framework of a multidisciplinary optimization. Material densities and maximum reuse temperature were computed. TPS mass was estimated assuming the category of materials used for the space shuttle and thickness distribution assigned on a review of HL-20 materials for each component. Trajectory-based TPS sizing has been proposed by Olynick [13] for a winged vehicle concept. The heating peak was determined considering an X-33 trajectory, discretized in a number of fixed waypoints. The resulting aerothermal database was used as an input for a one-dimensional conduction analysis, and several one-dimensional stackups of different materials representative of TPS were consequently sized. Bradford et al. [14] developed an engineering software tool for aero-heating analysis and TPS sizing. The tool is applicable in the conceptual design phase for reusable, non-ablative TPS. The thermal model was based on a one-dimensional analysis, and TPS was modeled considering a stackup of ten different material layers. Mazzaracchio [15] proposed a method to perform the sizing of a TPS depending on the locations of ablative and reusable zone on a TPS considering the coupling between trajectory and heat shield. Multidisciplinary analysis, integrating a procedural NURBS-based shape representation, is adopted for a preliminary design [3]. NURBS parameterization allows a simple control over the aerodynamic shape using a limited number of sensitive

However, derivation of a unique parameterization to describe the overall changes of geometry resulting from a shape optimization is not always possible, and several surfaces are used to parameterize different parts of the geometry. Implicit surfaces are a powerful and alternative tool for creating shapes due to their smooth blending properties enabling creation of arbitrary shape. In the present work, a soft object-derived representation for TPS thickness and material attribution is introduced. According to the legacy formulation of this technique, originally developed in computer graphics for the rendering of complex organic shapes [16], threedimensional object surfaces are (implicitly) obtained by defining a set of source points (or even more complex varieties) irradiating a potential field that is subsequently tracked according to an assigned isosurface. Following a quite different paradigm developed in [17], the full potential field irradiated by a set of bydimensional soft objects is congruently mapped on a discretized RLV shape. The methodology is able to create arbitrary TPS distributions seamlessly increasing

design parameters acting as geometrical modifiers.

materials [12].

16

Soft objects constitute a modeling technique which typically represents a domain using a scalar field, namely, a field function F, defined over a three-dimensional space. An implicit surface S defined as

$$\mathcal{S} = \left\{ \mathfrak{x} \in \mathbb{R}^3 | F(\mathfrak{x}) = T \right\} \tag{1}$$

that is, an isosurface S of the field function F specified by the threshold T represents an object instance using a raster conversion algorithm. Soft object modeling overcomes the drawback given by the parametric surfaces; that is, they automatically allow a self-blending between different primitives. Therefore, complex shapes can be modeled defining n ≥ 1 potential field fi, with origin in points xi, and the blending among them is formally accounted by the algebraic summation of their potential fields fi [18]:

$$F(d) = \sum\_{i=1}^{n} \hat{f}\_i(d\_i) \tag{2}$$

A commonly adopted notation

$$F\_i(d) = f\_i \circ d\_i \tag{3}$$

composes the distance metric di (which determines the shape of the objects associated to the key point xi), with the field function fi, being x the point of space in which the function is evaluated:

$$d\_i = \frac{||\mathbf{x} - \mathbf{x}\_i||\_k}{r\_i} \tag{4}$$

A more powerful representation used in soft object modeling is based on morphological skeleton that synthesizes the morphological properties of a given domain. A skeleton Sk can be defined as a basic geometric entity (such as points, segments, and plain closed domains) around which more complex shapes can be created once the distance function is provided. The simplest soft object was introduced by Blinn that originally proposed the "blobby molecule," an isotropically decaying Gaussian function modulated in strength and radius [16]:

$$f(d\_i) = \exp\left(-\frac{d\_i^2}{2}\right) \tag{5}$$

where d is the Euclidean distance (k = 2 in Eq. (4)). Blobby molecule is a soft object defined around a point skeleton, and its field function has an infinite support. This aspect affects the computational effort in a practical implementation, because it has to be evaluated in all points of the space. However, in literature, several finite support potential functions have been proposed for different modeling purposes. Wyvill et al. [19] developed the following field function:

$$f(d) = \begin{cases} 1 - \frac{22}{9}d^2 + \frac{17}{9}d^4 - \frac{4}{9}d^6 & d^2 < 1 \\\\ 0 & \text{otherwise} \end{cases} \tag{6}$$

The field function fi used in the present work has a finite support and assumes

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

Two-dimensional soft objects preserve self-blending property. Figure 1a, b shows the support and the strength field, respectively, created superposing n = 6 discrete point source blobs with radius r, with origins in key point xi. If δe < 2r, two or more blobs superposes, and the strength of the potential field is obtained summing up the strengths of each blob (see Figure 1b). A set of n blobs represents a too complex entity if used to model a parametric variation of shape (a single blob is characterized by five independent parameters, i.e., scalar coordinates of centers, strength, and radius). Therefore, blobs can be conveniently and easily arranged in macroaggregates with key points placed on a geometric segment (straight or curved) denoted from now on as "sticks." The point source blobs emulates a segment skeleton with the distance function expressed by Eq. (4) (see Figure 1a). However, a simple algebraic summation of potential fields creates a stick support having "bulges." Increasing the number of blobs, the shape of the support becomes

Stick primitives obtained with nblob = 6 and 20: constant radius (a, b); variable radius (c, d). The stick support becomes more regular increasing nblob; the strength field remains bounded to unit value.

arctan pð Þ � 2pd arctanp 0

d< 1 d≥ 1

(8)

normalized values in the range between 0 and 1 [18]:

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

f dð Þ¼

Figure 2.

19

8 < :

2.1 Two-dimensional integral soft object for TPS modeling

Blanc [18] proposed another field function introducing an internal hardness factor p, which tunes the blending between two different blobs. A higher value of p makes a blob stiffer in the blending, while a low hardness factor generates larger rounded shapes [17]:

$$f(d) = \begin{cases} 1 - \frac{9d^4}{p + (9/2 - 4p)d^2} d^2 \le 1/4 \\\\ \frac{(1 - d^2)^2}{3/4 - p + (3/2 + 4p)d^2} 1/4 < d^2 \le 1 \end{cases} \tag{7}$$

Figure 1. Support (a) and strength field (b) of a stick created by a superposition of n = 6 point source blobs.

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

The field function fi used in the present work has a finite support and assumes normalized values in the range between 0 and 1 [18]:

$$f(d) = \begin{cases} \frac{1}{2} + \frac{1}{2} \frac{\arctan(p - 2pd)}{\arctan p} d < 1\\ 0 \end{cases} \tag{8}$$

#### 2.1 Two-dimensional integral soft object for TPS modeling

Two-dimensional soft objects preserve self-blending property. Figure 1a, b shows the support and the strength field, respectively, created superposing n = 6 discrete point source blobs with radius r, with origins in key point xi. If δe < 2r, two or more blobs superposes, and the strength of the potential field is obtained summing up the strengths of each blob (see Figure 1b). A set of n blobs represents a too complex entity if used to model a parametric variation of shape (a single blob is characterized by five independent parameters, i.e., scalar coordinates of centers, strength, and radius). Therefore, blobs can be conveniently and easily arranged in macroaggregates with key points placed on a geometric segment (straight or curved) denoted from now on as "sticks." The point source blobs emulates a segment skeleton with the distance function expressed by Eq. (4) (see Figure 1a). However, a simple algebraic summation of potential fields creates a stick support having "bulges." Increasing the number of blobs, the shape of the support becomes

Figure 2.

Stick primitives obtained with nblob = 6 and 20: constant radius (a, b); variable radius (c, d). The stick support becomes more regular increasing nblob; the strength field remains bounded to unit value.

This aspect affects the computational effort in a practical implementation, because it has to be evaluated in all points of the space. However, in literature, several finite support potential functions have been proposed for different modeling purposes.

> 17 <sup>9</sup> <sup>d</sup><sup>4</sup> � <sup>4</sup>

Blanc [18] proposed another field function introducing an internal hardness factor p, which tunes the blending between two different blobs. A higher value of p makes a blob stiffer in the blending, while a low hardness factor generates larger

<sup>1</sup> � <sup>d</sup><sup>2</sup> � �<sup>2</sup>

Support (a) and strength field (b) of a stick created by a superposition of n = 6 point source blobs.

<sup>p</sup> <sup>þ</sup> ð Þ <sup>9</sup>=<sup>2</sup> � <sup>4</sup><sup>p</sup> <sup>d</sup><sup>2</sup> <sup>d</sup><sup>2</sup> <sup>≤</sup>1=<sup>4</sup>

<sup>3</sup>=<sup>4</sup> � <sup>p</sup> <sup>þ</sup> ð Þ <sup>3</sup>=<sup>2</sup> <sup>þ</sup> <sup>4</sup><sup>p</sup> <sup>d</sup><sup>2</sup> <sup>1</sup>=4<d<sup>2</sup> <sup>≤</sup><sup>1</sup>

<sup>9</sup> <sup>d</sup><sup>6</sup> <sup>d</sup><sup>2</sup> <sup>&</sup>lt; <sup>1</sup>

(6)

(7)

0 otherwise

Wyvill et al. [19] developed the following field function:

Hypersonic Vehicles - Past, Present and Future Developments

8 ><

>:

8 >>>>><

>>>>>:

<sup>1</sup> � <sup>22</sup>

<sup>9</sup> <sup>d</sup><sup>2</sup> <sup>þ</sup>

<sup>1</sup> � <sup>9</sup>d<sup>4</sup>

f dð Þ¼

f dð Þ¼

rounded shapes [17]:

Figure 1.

18

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 relation:

$$F\_j(P) = \max\_{\forall P} \left( F\_{j-1}(P), G\_j(P) \right) j = \mathbf{1}, \dots, n\_{\text{blobs}} \tag{9}$$

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:

$$G\_j(P) = \begin{cases} f(P) & d < r \\ 0 & \text{otherwise} \end{cases} \tag{10}$$

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

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

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

on the B-grid is shown in Figure 3.

Morphological (left) vs. topological map (right).

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

Figure 3.

21

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 varying field.
