T.E. Tezduyar, S. Sathe, J. Pausewang, M. Schwaab, J. Crabtree and J. Christopher, “Fluid–Structure Interaction Modeling with Moving-Mesh Techniques”, Proceedings of the Symposium on Recent Progress in Computational Fluid Dynamics, Japan Society of Automotive Engineers, Tokyo, Japan, 2007.
Fluid–Structure Interaction Modeling with Moving-Mesh Techniques* Tayfun E. TEZDUYAR1), Sunil SATHE1), Jason PAUSEWANG1), Matthew SCHWAAB1), Jason CRABTREE2), Jason CHRISTOPHER1)
The stabilized space–time fluid–structure interaction (SSTFSI) technique [1], developed by the Team for Advanced Flow Simulation and Modeling (T*AFSM), is a moving-mesh technique. It was applied in [1-3] to several 3D examples, including arterial fluid mechanics [2]. Here we focus on some of the supplementary techniques the T*AFSM developed to address various computational challenges involved in FSI problems, including problems with air–fabric interaction. The supplementary techniques include using split nodal values for pressure at the edges of the fabric and incompatible meshes at the air–fabric interfaces, the FSI Geometric Smoothing Technique, and the Homogenized Modeling of Geometric Porosity.
Keywords: Fluid–structure interaction, Space–time FSI technique, Air–fabric interaction 1. INTRODUCTION The stabilized space–time FSI (SSTFSI) technique was introduced in [1] to increase the scope and performance of the earlier space–time FSI techniques developed by the Team for Advanced Flow Simulation and Modeling (T*AFSM). The stabilization methods used in these techniques are the Streamline-Upwind/PetrovGalerkin (SUPG) [4] and Pressure-Stabilizing/PetrovGalerkin (PSPG) [5] formulations. A number of 3D examples computed with the SSTFSI techniques were presented in [1], and additional examples in [2,3], including examples from arterial fluid mechanics [2]. The SSTFSI technique is a moving-mesh (interfacetracking) technique, where the mesh moves to track (follow) the interface between the fluid and structure. Compared to non-moving mesh (interface-capturing) techniques, moving-mesh techniques result in more accurate representation of the flow field near the fluid– structure interface. In this paper, we focus on some of the supplementary techniques the T*AFSM developed to address various computational challenges involved in FSI problems, including problems with air–fabric interaction. With these supplementary techniques, moving-mesh techniques can still be used effectively in FSI computations that one would normally think of as far too complex for moving-mesh techniques. The versions of the SSTFSI technique we use in the computations presented here are the SSTFSI-TIP1, which was described in Remarks 5 and 10 in [1], and SSTFSISV, which was described in Remarks 6 and 10. In the air– * Presented at 2007 JSAE Annual Congress. 1) Mechanical Engineering, Rice University - MS 321 (6100 Main Street, Houston, TX 77005, USA) E-mail:
[email protected] 2) Civil and Mechanical Engineering, US Military Academy (West Point, NY 10996, USA)
fabric interaction computations, the fabric is modeled with the membrane element, which does not offer bending stiffness, and is assumed to be made of linearly elastic material. Air–fabric interactions pose a number of numerical challenges beyond those posed by an FSI problem in general. These additional challenges include stabilizing the structural response at the edges of the membrane structures without the benefit of any bending stiffness and sheltering the fluid mechanics mesh from the consequences of the geometric complexity of the structure. Two of the supplementary techniques we use in conjunction with the SSTFSI techniques help us stabilize the structural response at the edges and prevent excessive bending. The first one, proposed in Remark 9 in [1], is using split nodal values for pressure not only in the interiors but also at the boundaries (i.e. edges) of a membrane structure submerged in the fluid. Our earlier experience shows that this provides additional numerical stability for the edges of the membrane. The second one, proposed in [3], is using incompatible meshes at the fluid– structure interface as a means for limiting the excessive bending to narrow regions near the edges. This is accomplished by increasing the structural-mesh refinement near the edges without an equal refinement increase for the fluid mesh in the same regions of the fluid–structure interface. Sheltering the fluid mechanics mesh from the consequences of the geometric complexity of the structure is accomplished with the FSI Geometric Smoothing Technique (FSI-GST) and the Homogenized Modeling of Geometric Porosity (HMGP). The FSI-GST and HMGP are described in Sections 2 and 3. The HMGP was introduced to address one of the challenges involved in the FSI modeling of the new parachutes to be used with NASA's Orion space vehicle. We present our test computations in Section 4, Orion-parachute computations in Section 5, and concluding remarks in Section 6.
2.
FSI GEOMETRIC SMOOTHING TECHNIQUE The FSI Geometric Smoothing Technique (FSI-GST) was proposed in [1] for computations where the geometric complexity of the structure would require a fluid mechanics mesh that is not affordable or not desirable or just not manageable in mesh moving. In this technique, the structural mesh and displacement rates at the interface are projected to the fluid mesh after a geometric smoothing. In the geometric smoothing, a value (mesh coordinate or displacement rate) at a given node is replaced by a weighted average of the values at that node and a limited set of nearby nodes. When projecting the stress values from the smoothened interface to the structure, currently those values are just transferred to the corresponding nodes of the structure. In some computations, one may need not an isotropic geometric smoothing but a directional smoothing along some preferred direction. For such computations, the FSI Directional Geometric Smoothing Technique (FSI-DGST) was proposed in [1] as a version of the FSI-GST. In the FSI-DGST, whenever possible, the interface mesh is generated in such a fashion that the preferred smoothing directions can approximately be represented by the gridlines of the interface mesh. Then the weighted averaging for a node on such a gridline would involve a limited set of nearby nodes only along that gridline. The directional smoothing concept is similar to the directional "upwind" concept of the SUPG formulation, where the residual-based numerical dissipation is active only along the streamline direction. In a special version of the FSI-DGST introduced in [6] for parachutes, the smoothing is carried out in the circumferential direction of the parachute canopy. This addresses the geometric complexities associated with the "peaks" and "valleys" of the parachute gores, which are formed by the inflation of a canopy with embedded reinforcement cables positioned longitudinally in the canopy structure. In this FSI-DGST for parachutes, certain nodes from the structure interface mesh are picked to generate the set of fluid interface nodes. Therefore the number of fluid interface nodes are smaller than the number of structure interface nodes. While generating the set of fluid interface nodes, the structure interface nodes from the valleys are picked. In picking these nodes circumferentially, a few valleys can be skipped, and in picking them longitudinally, inside a valley a few nodes can be skipped. The nodes are then connected with threenode triangular elements, resulting in a smooth fluid mesh along the circumferential direction. For parachutes with large number of gores (80 gores in the ringsail parachute studied in this paper), the distance by which the gores bulge out is small compared to the parachute diameter. It was postulated in [6] that keeping the true shapes of the gores in the flow computations is not essential for calculating the fluid dynamics forces in this class of applications. A value (mesh coordinate or displacement rate) at a given fluid interface node is replaced by the value at the mapping structure interface node. When transferring the stress values from the smoothened interface to the structure, the values for the mapping nodes
are directly transferred, and for the remaining nodes a weighted average is used. 3.
HOMOGENIZED MODELING OF GEOMETRIC POROSITY The Homogenized Modeling of Geometric Porosity (HMGP) was introduced in [6] to address one of the computational challenges involved in the FSI modeling of the new parachutes to be used with NASA's Orion space vehicle. That particular challenge is the geometric porosity of the parachute canopy, a consequence of the many "rings" and "sails" used in constructing the parachute canopy. In the STTFSI techniques introduced in [1] the fabric porosity is accounted for (see Eq. (28) in [1]), and one of test problems reported in [1] was the descent of a T–10 parachute with a realistic fabric porosity. With the HMGP, the intractable complexities of the geometric porosity is bypassed by approximating it with an "equivalent", locally-varying fabric porosity. The fluid mechanics computations see a "solid" parachute, where the slits between the rings and sails are filled with fabric. The structural mechanics computations retain the rings and sails construction of the parachute canopy. The solid parachute is assigned a locally-varying fabric porosity so that it closely approximates the parachute with slits. The equivalent fabric-porosity coefficient is calculated from the cross-canopy pressure differentials and flow rates computed for the parachute with geometric porosity. This involves a one-time flow computation for the parachute with geometric porosity. From this, one can calculate a locally-varying porosity coefficient, or, as a special case of that, a single, uniform porosity coefficient. After calculating the locally-varying porosity coefficient, it is calibrated by scaling it up or down to match the expected drag. In our parachute computations the FSIDGST and HMGP steps are combined into a single step. 4. TEST COMPUTATIONS All computations reported in this section are carried out in a parallel computing environment, using PC clusters. The meshes are generated on a single node of the cluster used. All computations were completed without any remeshing. In all cases, the fully-discretized, coupled fluid and structural mechanics and mesh-moving equations were solved with the quasi-direct coupling technique (see Section 5.2 in [1]). In solving the linear equation systems involved at every nonlinear iteration, the GMRES search technique [7] was used with a diagonal preconditioner. 4.1 Windsock The windsock model in our computation has a length of 1.5 m and a diameter ranging from 0.25 m upstream to 0.15 m downstream. Initially the windsock is in a horizontal position, and the starting condition for the flow field is the developed flow field corresponding to a rigid windsock held in that horizontal position. Then the gravity is turned on for the windsock, the FSI starts, and the windsock starts bending down. The wind velocity is constant at 10 m/s. The thickness, density, stiffness and Poisson's ratio for the windsock are 2.0 mm, 100 kg/m3,
1.0×106 N/m2 and 0.45, respectively. The upstream edge of the structure is held fixed while the remaining structure is free and flaps in cycles. We use the FSI-GST for smoothing the fluid mesh at the interface. The nodes of the windsock mesh are generated on straight longitudinal gridlines, and with that we are able to use the directional version of the FSI-GST, i.e. the FSI-DGST. For a node on such a gridline, we use a weighted averaging involving four nearby nodes on each side (see Section 12.4 in [1]). This directional smoothing does not introduce smoothing in the circumferential direction. During the FSI computations the structure develops kinks, which would make updating the fluid mechanics mesh difficult and increase the remeshing frequency. With the FSI-DGST, two flapping cycles were computed without remeshing. Fig. 1 shows the structure and fluid meshes at the interface, one with a kink and the other one smooth. Fig. 2 shows the windsock and the flow field at four instants. For more details on this computation, see [1].
respectively. The diameter, density, stiffness and Poisson's ratio for the cables are 6.35 mm, 1,440 kg/m3, 3.0×109 N/m2 and 0.3, respectively. Fig. 3 also shows the fluid mechanics mesh at the interface, the sail meshes (which are more refined than the fluid mechanics mesh at the interface), and the sail at three instants. For more details on this computation, see [3,6].
Figure 1. Windsock. Meshes at the interface: structure (left) and fluid (right).
Figure 2. The windsock and the flow field (velocity and pressure) at four instants. 4.2 Sails Fig. 3 shows the dual-sail configuration. The mainsail geometry and initial shape are derived from a plate airfoil design with maximum camber of 15% cord length located at 30% cord. This is an approximation to the mainsail of the Adventuress [8]. The airflow velocity is 7.72 m/s at an angle of 35° from the centerline of the boat (not modeled). The thickness, density, stiffness and Poisson's ratio for the sails are 1.0 mm, 1,370 kg/m3, 3.0×109 N/m2 and 0.3,
Figure 3. Sails. Configuration (top-left), fluid mechanics mesh at the interface (top-right), sail meshes (middle-left), and the sails at three instants.
5.
RINGSAIL PARACHUTE FOR THE ORION SPACE VEHICLE We assume that NASA will be using a cluster of three ringsail parachutes during the terminal descent of the Orion space vehicle. The three ringsail parachutes, referred to as the "mains'', are being designed to support a weight of approximately 15,000 lbs at a steady descent speed of 25 ft/s. To better understand the performance of the mains, we are currently modeling a single main under different conditions. 5.1 Parachute Components, Geometry and Material The ringsail parachutes are made of several rings or sails and are ogival in their unstressed shape. The ringsail parachute studied here has a profile of a quarter of a sphere in its unstressed shape. The crown portion of the ringsail parachute (the portion near the vent) is made of rings with gaps between the consecutive rings (see Fig. 4). The middle and skirt portions of the parachute are made of sails. Two edges of the sails are stitched to the radial lines and the other two edges are free. The edge facing the parachute skirt is called the leading edge and the edge facing the vent is called the trailing edge. The leading and the trailing edges could have fullnesses so that they appear bulged out even in the unstressed state. The canopy construction includes several bands, lines and tapes that provide structural stiffness. The vent band provides the necessary strength to the vent, where the stress concentration is high. The radial lines provide stiffness along the longitudinal direction and cause the formation of the gores in the parachute. The skirt band, which connects the ends of the leading edges of the last sail in each gore, is often used for controlling the opening of the parachute. Individual sails or rings are sometimes reinforced with tapes on the leading and trailing edges to prevent tearing. The suspension lines connect the skirt end of each radial line to a single riser connected to the payload.
Figure 4. Top: Gore layout for the ringsail parachute (not drawn to scale). Middle and Bottom: Four-gore structure and fluid meshes at the interface.
The ringsail parachute here has 80 gores and a nominal diameter of about 120 ft. It has 4 rings and 9 sails, and together they form a quarter of a spherical surface. The rings and sails are shown in Fig. 4, where a single gore is laid out flat. The fullness values for the sails were provided to us by NASA JSC. The suspension lines are about 130 ft in length and the riser is about 25 ft. The payload is about 5,000 lbs, represented by a point mass. The canopy of the ringsail is made of different materials. The material properties for the rings and sails were provided to us by NASA JSC. The ringsail parachute modeled here includes radial lines, suspension lines, risers, a vent band, a skirt band, and leading- and trailingedge tapes. The material properties for these components of the ringsail were also provided to us by NASA JSC. 5.2 Smoothing and Homogenization We use incompatible meshes at the fluid–structure interface. The structure mesh is very refined and models each individual ring, sail and gore of the parachute. The fluid mesh at the interface is coarser. We use the FSI-GST described in Section 2 to generate and update the fluid mesh at the interface. The vent for the ringsail is very small, and keeping one element per gore would have resulted in extreme mesh refinements. Therefore, in the circumferential direction, for the rings we pick every other valley node, and for the sails every valley node. To keep the element aspect ratios reasonable, in the longitudinal direction, for the first ring we pick every other valley node. For the second ring we pick three valley nodes, and for each of the remaining rings and sails we pick two valley nodes. Fig. 4 shows, for four gores, the structure and fluid meshes at the interface. The fluid mesh is sufficiently refined but has significantly less nodes. We use two homogenized models for geometric porosity. In the first model the geometric porosity is represented by a single, uniform porosity for the entire canopy. A porosity coefficient of 262.6 CFM gives us the expected drag. A good portion of the results reported in [6] were obtained with this uniform porosity. We note that these results meet out expectations, and therefore we believe that the model has acceptable accuracy and is attractive because of its simplicity. In the second model, we represent the geometric porosity with a locally-varying fabric porosity. We divide the canopy into 12 concentric patches and calculate an equivalent fabric-porosity coefficient for each. Each patch includes a slit, and part of a ring or sail on either side of the slit. Patch 1 includes the first ring completely, and Patch 12 includes the last sail completely. Fig. 5 shows Patch 4 of the four-gore slices of the fluid and structure interfaces.
Figure 5. Patch 4 of the four-gore slice of the fluid (left) and structure (right) interfaces.
We carry out a one-time flow computation for a fourgore canopy slice with all the rings, sails and slits to calculate a porosity coefficient for each patch. Using only a four-gore slice keeps the problem size at a manageable level. The four-gore fluid surface is held rigid and the free-stream velocity is set to 25 ft/s. The flow computation is carried out until a fully-developed flow is reached. Fig. 6 shows the flow field. The porosity coefficient for each patch can be calculated based the cross-canopy pressure differential and flow rate associated with that patch (for details, see [6]). We calibrate this porosity coefficient so that the parachute with the homogenized porosity generates the expected drag of approximately 5,000 lbs. Scaling up the porosity coefficients by a factor of 2 yields the expected drag. We therefore use this calibration factor in computations with locally-varying fabric porosity. Fig. 7 shows the smoothened, homogenized fluid interface colored by the porosity coefficient.
5.3 Computational Parameters All conditions described in the introductory part of Section 4 are also applicable in this section. The meshes are partitioned to enhance the parallel efficiency of the computations. Mesh partitioning is based on the METIS [9] algorithm. For additional information on the computational parameters, see [6]. 5.4 Offloading It is anticipated that the Orion space vehicle will need to reduce its descent speed just before landing. Its estimated descent speed of 25 ft/s may not be low enough for landing. This would especially be the case if NASA decides to recover the space vehicle on land as opposed to ocean like they did in the Apollo program. One way to achieve reduced descent speeds at landing is to reduce the weight of the vehicle, popularly known as "offloading". Using our general FSI techniques described in [1] and more specialized techniques described in this paper, we investigate the effects of offloading. The heat shield is approximately 13.5% of the weight of the vehicle and is the most likely candidate for offloading. We computed this case with the locally-varying porosity model. Fig. 8 shows the parachute shape before and about 6 s after the heat shield is dropped. Fig. 9 shows the flow field before and about 6 s after the heat shield is dropped.
Figure 6. Flow field for the four-gore canopy slice.
Figure 7. Fluid interface colored by the porosity values.
Figure 8. Parachute shape before and about 6 s after the heat shield is dropped.
computed the air–fabric interactions of a windsock, a pair of sails and the Orion parachute. The results demonstrate that the SSTFSI technique, together with the supplementary techniques we described, can successfully address the numerical challenges involved in real-world FSI problems, including those with air–fabric interactions. ACKNOWLEDGEMENTS This work was supported in part by NASA JSC under Grant NNJ06HG84G and by the Rice Computational Research Cluster funded by NSF under Grant CNS0421109, and a partnership between Rice University, AMD and Cray.
Figure 9. Flow field before and about 6 s after the heat shield is dropped. The velocity vectors are colored by magnitude. 6. CONCLUDING REMARKS In this paper we focused on some of the supplementary techniques the T*AFSM has developed to be used in FSI computations with moving-mesh techniques. These supplementary techniques address various computational challenges involved in FSI problems, including problems with air–fabric interaction. With these supplementary techniques, moving-mesh FSI techniques, such the SSTFSI technique, can still be used effectively in computations that one would normally think of as too complex for moving-mesh techniques. Two of the supplementary techniques help us stabilize the structural response at the edges and prevent excessive bending. The first one is using split nodal values for pressure not only in the interiors but also at the boundaries of a membrane structure submerged in the fluid. The second one is using incompatible meshes at the fluid– structure interface as a means for limiting the excessive bending to narrow regions near the edges. The other two supplementary techniques shelter the fluid mechanics mesh from the consequences of the geometric complexity of the structure. These are the FSI Geometric Smoothing Technique (FSI-GST) and the Homogenized Modeling of Geometric Porosity (HMGP). As numerical examples, we
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