1 Introduction

This is the second part of a two-part article on T-splines computational 2D–1D structural mechanics, where structures with different parametric dimensions are connected with continuity and smoothness. In the first part [1], the method and its implementation were presented, referring to the 2D structure as “membrane” and the 1D structure as “cable.” The method and its implementation are of course applicable also to other 2D–1D cases, and the test computations presented in the first part were not only for membrane–cable structures but also for shell–cable structures. In the second part, we are presenting spacecraft parachute structural mechanics computations with the method introduced in the first part. The computations are with both the membrane and shell models of the parachute canopy fabric. The computer model used in the computations is for a subscale, wind-tunnel version of the Disk–Gap–Band (DGB) parachute.

Table 1 Material and cross-sectional properties for the membrane and shell. Young’s modulus E, thickness \(h_\textrm{th}\), density \(\rho \), and Poisson’s ratio \(\nu \)
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Table 2 Material and cross-sectional properties for the bending-stabilized cable. Young’s modulus E, area A, density \(\rho \), and second area moment \(I_0\)
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The basis functions derived in the method development presented in the first part give us the desired smoothness between structures with 2D and 1D parametric dimensions. The derivation involves proper selection of a scale factor for the knot vector of the 1D structure and results in new control-point locations. While the method description in the first part focused on \(C^0\) and \(C^1\) continuity, paths to higher-order continuity were marked where needed. When the membrane and cable are connected with smoothness, the strain and rotational freedoms are transferred between the two structures. For easy and efficient implementation of the method, in the first part, the Beziér extraction row operators were introduced to be used in obtaining the basis functions. As the limitations of the method, two points were mentioned in the first part. The smoothness can only be achieved along the cable and its parametric-line continuation in the membrane. The starting membrane and cable meshes need to meet some additional requirements if they are to be connected with a continuity higher than \(C^1\).

A large number of membrane–cable structural mechanics computations were reported in the journal articles published by the second and fourth authors on parachutes [2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24], going as far back as the year 2000 [2, 3] and focusing on spacecraft parachutes starting in 2008 [7]. These were essentially all part of parachute fluid–structure interaction (FSI) computations with the space–time (ST) computational methods [6, 7, 25,26,27,28,29]. They were all parallel computations, all essentially with the parallel implementation given in [2]. Except for the computations in [22, 24], which were with isogeometric discretization, they were all with finite element discretization, based on various versions of the membrane and cable structural mechanics formulations given in [30, 31]. The membrane meshes for the spacecraft parachutes were essentially all made of quadrilateral elements.

Fig. 1
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DGB parachute configuration

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The first parachute canopy NURBS mesh created in connection with the ST FSI computations was for gore curvature calculations [19] from FSI modeling of a drogue parachute with finite element discretization. The structural displacements from the FSI computation were projected to the NURBS mesh used in the curvature calculations. The mesh was quadratic NURBS in the radial direction and cubic NURBS in the circumferential direction. Considering the continuity objectives in projecting the FSI solution to the NURBS mesh, the origins of the motivation and ideas behind the methods introduced in the first part of the article can be traced back to the work on gore curvature calculations.

The membrane–cable structural mechanics computation in [22] was for a ram-air parachute. The mesh was quadratic NURBS. The computations in [24] were for a drogue parachute, with the canopy constructed from ribbons. The canopy had 24 gores, each composed of 52 horizontal ribbons and each having seven parallel, equidistant vertical tapes retaining the ribbons. The gores were separated by tapes, called “radial lines,” continuing beyond the canopy as suspension lines, with the 24 suspension lines connecting to a single riser. The canopy also had tapes along the skirt edge and around the vent, called “skirt band” and “vent band.” The ribbons were modeled with cubic NURBS membrane elements, and all tapes, lines, and the riser were modeled with cubic NURBS cable elements. The point where a cable intersected a ribbon edge was a cable patch boundary. The cables separating the gores and the cables running across ribbons were membrane patch boundaries. All patches had \(C^0\) continuity between them.

Fig. 2
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Undeformed parachute. Brown lines indicate the tapes, suspension lines, risers, and extension line. (Color figure online)

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Table 3 Mesh-N, Mesh-T1, and Mesh-T2. Number of control points (nc) and the number of 2D and 1D elements (\(ne_{\textrm{2D}}\) and \(ne_{\textrm{1D}}\))
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In the DGB parachute computations here, we use three types of meshes in combination with the membrane and shell models of the canopy fabric. In total, we compute six cases. Although the DGB parachute has a different configuration than the drogue parachute computed in [24] had, the first one of the three meshes we use here is constructed conceptually the same way the drogue parachute NURBS mesh was constructed. The second and third meshes are constructed with the methods and ideas introduced in the first part of the article. We present the computations in Sect. 2 and the concluding remarks in Sect. 3.

2 Computations

2.1 Parachute description

The DGB parachute to be used in the supersonic flow regime in Mars exploration has a nominal diameter of 20 m. Our computer model is for a subscale version used in wind tunnel tests. It has a nominal diameter of \(D_\textrm{o} = 171\, {\textrm{mm}}\). Figure 1 shows the parachute configuration. It consists of the canopy, suspension lines, risers, and the extension line. The canopy is made of the disk, band, and gap tapes that go across the gap between the disk and the band. The disk and band have 24 gores made of canopy fabric, with canopy tapes between the gores. The vent tape goes around the vent, and the edge tapes go along the other disk and band edges. We consider two models for the canopy fabric, membrane with wrinkling model [31] and shell [32], and two models for the canopy tapes, bending-stabilized cable [33] and shell. All other tapes, the suspension lines, risers and the extension line are modeled as bending-stabilized cable. The disk has a diameter of 171 mm, the band has a width of 29 mm, and the gap has a width of 10 mm. The suspension lines are 279 mm long, the risers are 130 mm long, and the extension line is 49 mm long. The material and cross-sectional properties, shown in Tables 1 and 2, are from Fujikura Parachute Co., Ltd.

Fig. 3
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Mesh-N. The Brown lines indicate the 1D structures, and the checkerboard pattern is for differentiating between the 2D elements. (Color figure online)

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Fig. 4
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Mesh-N after the parachute inflation. Zoomed view of the elements and control points where a gap tape is connected to the disk skirt. The Brown control points are for representing the 1D structures, which are marked in Brown, and the Green control points are for representing the 2D structures, where the element edges are marked in Green. The Brown control points used in representing the vent, canopy, and edge tapes are also used in representing the 2D structures. (Color figure online)

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Fig. 5
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Mesh-T1. The Brown lines indicate the 1D structures, and the checkerboard pattern is for differentiating between the 2D elements. (Color figure online)

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Fig. 6
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Mesh-T1 after the parachute inflation. Zoomed view of the elements and control points where a gap tape is connected to the disk skirt. The Brown control points are for representing the 1D structures, which are marked in Brown, and the Green control points are for representing the 2D structures, where the element edges are marked in Green. The Brown control points used in representing the vent, canopy, and edge tapes are also used in representing the 2D structures, using the ideas and methods introduced in [34]. (Color figure online)

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Remark 1

We note that the original bending-stabilized cable model does not preclude compressive stress. The reduction in the compressive stress is due to the buckling facilitated by the bending stiffness. However, when the cable is embedded in the 2D surface, the buckling makes a significant impact on the membrane behavior. Therefore, by turning the cable slacking model “on,” we preclude the compressive stress.

Fig. 7
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Mesh-T2. The Brown lines indicate the 1D structures, the checkerboard pattern is for differentiating between the 2D elements, and the alternating Blue and Yellow pattern is for the strips of 2D elements representing the canopy tapes placed over the same radial strips of fabric elements over which the canopy tapes were placed in Mesh-T1. (Color figure online)

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Fig. 8
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Mesh-T2 after the parachute inflation. Zoomed view of the elements and control points where a gap tape is connected to the disk skirt. The Brown control points are for representing the 1D structures, which are marked in Brown, the Green control points are for representing the 2D fabric structures, where the element edges are marked in Green, and the Red control points are for representing the 2D canopy tape structures, where the element edges are marked in Red. The Brown control points used in representing the vent and edge tapes are also used in representing the 2D fabric and canopy tape structures. The Red control points used in representing the 2D canopy tape structures are also used in representing the 2D fabric structures. (Color figure online)

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Fig. 9
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Parachute after the inflation. Membrane Fabric. Mesh-N

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Fig. 10
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Parachute after the inflation. Membrane Fabric. Mesh-T1

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Fig. 11
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Parachute after the inflation. Membrane Fabric. Mesh-T2

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Fig. 12
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Parachute after the inflation. Shell Fabric. Mesh-N

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Fig. 13
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Parachute after the inflation. Shell Fabric. Mesh-T1

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Fig. 14
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Parachute after the inflation. Shell Fabric. Mesh-T2

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2.2 Problem setup

Figure 2 shows the undeformed parachute. In the structural mechanics computations we perform to obtain the inflated parachute shape, we use a uniform pressure difference of \(66.0\,{\textrm{kPa}}\) for the disk and \(39.6\, {\textrm{kPa}}\) for the band. We use three types of meshes, which we label “Mesh-N,” “Mesh-T1,” and “Mesh-T2.” In all three meshes, the basis functions are cubic NURBS or T-splines, and the gap tapes and suspension lines are connected to the disk and band with \(C^0\) continuity. The number of control points and elements in the three meshes are shown in Table 3.

Figure 3 shows Mesh-N. This is conceptually the same way the drogue parachute NURBS mesh was constructed in [24]. The mesh is made of NURBS patches for both 2D and 1D elements. The gores are NURBS patches, with \(C^0\) continuity between the gores, the canopy tapes are along the patch boundaries between the gores, and the vent tape and edges tapes are along the other boundaries of the 2D patches. The gap tapes and the suspension lines are connected to the corners of the 2D patches. Figure 4 shows, for the inflated parachute, zoomed view of the elements and control points where a gap tape is connected to the disk skirt.

Figure 5 shows Mesh-T1. For both the disk and the band, the fabric mesh starts as a circumferentially periodic B-splines patch, with \(C^2\) continuity also between the gores. The canopy tapes, separating the gores, are placed centrally along radial strips of elements. The representation of the canopy tapes over the 2D structures is based on the ideas introduced in [34]. The gap tapes and the suspension lines are connected to the 2D structures using the method introduced in [34]. Figure 6 shows, for the inflated parachute, zoomed view of the elements and control points where a gap tape is connected to the disk skirt.

Figure 7 shows Mesh-T2. The fabric elements are the same as in Mesh-T1. The canopy tapes, however, are represented not by 1D elements but by radial strips of 2D elements placed over the same radial strips of fabric elements over which the canopy tapes were placed in Mesh-T1. We use a shell model [32] for the 2D canopy tapes. Figure 8 shows, for the inflated parachute, zoomed view of the elements and control points where a gap tape is connected to the disk skirt.

2.3 Results

In modeling the canopy fabric, we will identify the membrane model with wrinkling as “Membrane Fabric,” and the shell model as “Shell Fabric.” We compute total six cases, which are the combinations of the two canopy fabric models with the three meshes.

Figures 9, 10, 11, 12, 13 and 14 show the parachute shape after the inflation. The difference between the shapes from the membrane and shell models is small, most discernible with Mesh-N. With the Membrane Fabric, there is wrinkling at the gore center at the disk skirt edge, but there is no such localized behavior with the Shell Fabric. There is no wrinkling with Mesh-T1 or Mesh-T2.

Figures 15 and 16 show the scaled mean curvature after the parachute inflation. The scaling is by \(D_\textrm{o}^{-1}\). The difference between the mean curvatures from the membrane and shell models is small. With Mesh-N, there is no negative mean curvature. However, theoretically, there is minus-infinity curvature where the canopy tapes are. With Mesh-T1, the curvature is negative where the tapes are; it is like bending there. With Mesh-T2, the curvature is almost zero where the tapes are, and the bending starts at the tape edges. This means that the bending stiffness coming from the tape is more than the stiffness coming from the discretization with \(C^2\) continuity. Based on these observations, because of the relatively wide tapes in the subscale parachute model, Mesh-N is not suitable for representing the canopy.

Fig. 15
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Canopy mean curvature, scaled by \(D_\textrm{o}^{-1}\), after the inflation. Membrane Fabric. Mesh-N, Mesh-T1, and Mesh-T2

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Fig. 16
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Canopy mean curvature, scaled by \(D_\textrm{o}^{-1}\), after the inflation. Shell Fabric. Mesh-N, Mesh-T1, and Mesh-T2

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3 Concluding remarks

This was the second part of a two-part article on T-splines computational 2D–1D structural mechanics, where structures with different parametric dimensions are connected with continuity and smoothness. In the first part, the method and its implementation were presented. The basis functions derived in the method development give us the desired smoothness between structures with 2D and 1D parametric dimensions. Although in the first part the 2D structure was referred to as “membrane” and the 1D structure as “cable,” the method and its implementation are of course applicable also to other 2D–1D cases, and the test computations presented in the first part included shell–cable structures. In the second part, we have presented spacecraft parachute structural mechanics computations with the method introduced in the first part. The computer model used in the computations was for a subscale, wind-tunnel version of the DGB parachute. The computations were with both the membrane and shell models of the parachute canopy fabric. We used three types of meshes in combination with the two models of the canopy fabric. In total, we computed six cases. Although the DGB parachute has a different configuration than a drogue parachute, the first one of the three meshes was constructed conceptually the same way the NURBS mesh was constructed in a drogue parachute computation carried out earlier. The second and third meshes were constructed with the methods and ideas introduced in the first part of the article. The computations presented show the effectiveness of the method in 2D–1D structural mechanics computation of spacecraft parachutes.