Challenges faced in computational analysis of aortic-valve to aorta flow include (i) contact between the valve leaflets, which changes the flow-domain topology, (ii) high-resolution boundary-layer representation even with the topology changes, (iii) thin leaflets that generate thin shear layers and associated vortex structures, and (iv) flow driven primarily by pressure differences. In addressing these challenges with the Space–Time Computational Flow Analysis (STCFA), we use the “ST-SI-TC-IGA.” The core component of the ST-SI-TC-IGA is the ST Variational Multiscale (ST-VMS) method, and the other key components are the ST Slip Interface (ST-SI) and ST Topology Change (ST-TC) methods and the ST Isogeometric Analysis (ST-IGA). The Constrained-Flow-Profile Traction is also a part of the STCFA here, enabling us to use traction boundary condition at the inlet. Furthermore, for more cost-effective mesh refinement near the solid surfaces of the complex geometries involved, we are using a moving T-splines mesh with the product-form T-splines basis functions. This is enabled by the Complex-Geometry T-Splines Mesh Generation method. The mesh motion is enabled by a combination of the Fiber-Reinforced Hyperelasticity Mesh Update Method and the “ST-C,” a method for temporal data representation, relying on IGA basis functions in time. Our objective in the computational flow analysis here is to clarify how the geometric features of the aortic arch influence the flow patterns around the leaflets of a bioprosthetic aortic valve. That will enable a better understanding of (a) the mechanisms by which the aortic curvature and torsion influence the near-leaflet flow patterns, such as flow reversal and vortex dynamics, and (b) the wall shear stress and oscillatory shear index distributions resulting from those flow patterns.

Abstract The Space–Time Computational Flow Analysis (STCFA) enabled the virtual test and evaluation of spacecraft parachutes and helped reach the final design of the parachutes used in the Orion spacecraft landing in April 2026. Computational fluid–structure interaction (FSI) analysis of spacecraft parachutes requires overcoming a number of formidable challenges. (i) Because the parachute dynamics involves very large air masses compared to the parachute structure mass, the coupled fluid and structure equations need to be solved with the most robust coupling method practically realizable. Otherwise, the computation can generate nonphysical parachute-shape fluctuations, which would be mistaken for actual, physical shape fluctuations. That would, of course, throw off the design process if the design objective is to minimize the physical fluctuations. (ii) Spacecraft parachute canopies have a “ringsail” construction, creating a “geometric porosity” with hundreds of ring gaps and sail slits that the flow goes through to enhance the canopy shape stability, making the FSI problem seemingly intractable. (iii) Spacecraft parachutes are typically used in cluster configurations, where the parachute canopies interact with each other, creating flow computation challenges associated with large mesh deformations and contact between moving structural surfaces. (iv) The final parachute design had, beyond the hundreds of gaps and slits, a “modified geometric porosity,” aimed at enhancing the canopy shape stability. In FSI computations with such modified geometric porosity, the flow through the “windows” created by the removal of the panels and the wider gaps created by the removal of the sails has to be represented in a different way than the flow through the gaps and slits, because they are larger and their sizes are design parameters. Together with the core STCFA methods, a number of ancillary methods played an essential role in overcoming these challenges, leading to the first-ever FSI computation of the Orion spacecraft final-design parachute clusters, which was reported in 2013. We provide an overview of the core and ancillary STCFA methods that enabled the computations reported in 2013 and summarize the computations carried out for various parachute designs tried before reaching the final design and for other types of parachutes involved in the deceleration of the Orion spacecraft as it returns to Earth. We also provide an overview of the methods we introduced since 2013 to increase the fidelity and scope of spacecraft parachute computational flow analysis and summarize the computations carried out since then. These new developments include extending the computations to compressible flows and upgrading the discretization from finite elements to isogeometric analysis, first with NURBS and then with T-splines.

In the space-time (ST) computational analysis, the discretization methods, such as those with finite elements, have appeared in a number of variations over the past years. Most of the current methods employ discontinuous functions in time. The motivation for that is to avoid a cost associated with fully 4D computations. Consequently, the increase in the number of unknowns in time remains under control despite using ST elements. Such methods have been applied over the past decades to a very large number of fluid dynamics and advection-diffusion problems, which are governed by equations with first-order time derivatives, and to a far lesser extent to elastodynamics and other solid mechanics problems, which are governed by equations with second-order time derivatives. In both contexts, especially when the functions are continuous in space, the ST methods are often formulated in the framework of stabilized methods, such as the Streamline-Upwind/Petrov-Galerkin, Galerkin/Least-Squares, Pressure-Stabilizing/Petrov-Galerkin, and variational multiscale methods. These are sometimes supplemented with discontinuity-capturing techniques. With the widespread adoption of isogeometric analysis (IGA), the ST methods have also been synthesized with IGA and applied to a large number of problems, mostly in fluid dynamics. The synthesis, ST-IGA, enables and encourages the use of higher-order functions in time. Within the ST domain, increasing the polynomial order in time leads to the expected improvement in solution accuracy. At the lateral boundaries with Dirichlet condition, however, the fluxes do not gain increased accuracy with higher-order polynomials in time. Motivated by this concern, we focus here on 1D elastodynamics and how to treat the Dirichlet boundaries. We introduce two new stabilization methods and show, with test computations, how the stabilized equations used in computing the fluxes perform. The methods are consistent in the way the fluxes are computed and can be extended to 2D and 3D problems.

We present a cable isogeometric analysis method based on the Kirchhoff–Love beam theory coupled with Coulomb friction. In simulation of cable entanglement, twist plays a key role. Isogeometric rotation-free bending-stabilized cable formulations are attractive in that they can incorporate bending effects without introducing additional rotational degrees of freedom. However, since the cross-sectional orientation is entirely determined by the centerline geometry, such formulations inherently lack torsional degrees of freedom. As a result, they are unable to represent twisting behavior and cannot properly account for cross-sectional orientation, and that limits their applicability to problems involving pre-bent configurations or geometries with non-axisymmetric cross sections. This limitation is particularly significant in the presence of frictional contact, where tangential forces acting on the cable surface naturally induce torsional effects. Therefore, a formulation that can account for torsion is essential for accurately capturing the resulting mechanical response. One possible approach in incorporating torsional effects is to employ beam models with rotational degrees of freedom, such as the Kirchhoff–Love beam model, which has been widely used in classical finite element formulations. Isogeometric implementations of such beam models have also been proposed in the literature. In this work, rather than introducing additional complexity into the beam formulation itself, we revisit the classical Kirchhoff–Love theory and provide a concise and consistent formulation within the isogeometric framework. Our focus is on the coupling with Coulomb friction, enabling an accurate representation of contact behavior with torsional effects. We present test computations with contact between a rigid body and an inclined rigid plane and between a flexible cable and an inclined rigid cable. In both problems, the tests cover the frictionless, stick, and slip cases. The results show the accuracy and robustness of the cable isogeometric analysis method presented.

We present high-resolution Space–Time Isogeometric Analysis (ST-IGA) of NREL 5MW wind turbine long-wake flow, computed up to 10 rotor diameters downstream of the turbine. The ST Variational Multiscale (ST-VMS) method serves as the core method in the computation. The time-periodic velocity data at the inflow boundary of the wake domain comes from a wind turbine rotor and tower aerodynamics computation conducted earlier with the ST-IGA and ST-VMS. The wake flow is computed with the Carrier-Domain Method (CDM), introduced for high-resolution, high-efficiency computation of time-periodic long-wake flows. In the CDM, a short segment of the wake domain, the carrier domain (CD), moves in the free-stream direction, from the beginning of the long wake domain to the end. The data at the moving inflow plane comes from the time-periodic data computed at an earlier position of the CD. With the high mesh resolution that can easily be afforded over the short domain segment, the wake flow patterns can be carried, with superior accuracy, far downstream. The CDM has two versions, one where the CD moves in a continuous fashion (“CDM-C”), and one where it moves in a discrete fashion (“CDM-D”). The computations here are with the CDM-D. First, as a test long-wake flow computation with the CDM-D, we compute the 2D wake flow for a cylinder, at Reynolds number 100, up to 350 diameters downstream of the cylinder. We show that the wake flow is nearly indistinguishable from what is computed over the full wake domain (FWD). Next, we compute the wind turbine wake up to 5 rotor diameters downstream, showing again a very good match with the wake computed over the FWD. Following that, we extend the wake computation up to 10 diameters downstream. The computations presented demonstrate that the ST-IGA, ST-VMS, and CDM form a powerful computational framework for wind turbine long-wake flow analysis.

We present a T-splines mesh generation method for complex geometries, enabling good local mesh refinement in isogeometric analysis (IGA), with the continuity and smoothness desired. Good local mesh refinement enhances the IGA computational efficiency by having a higher density of control points only in places and directions where we need higher refinement. It also enhances computational robustness by evading high-aspect-ratio elements associated with directional refinement of structured meshes. The method is based on converting a multipatch NURBS mesh created by a complex-geometry mesh generation method to a product-form-T-splines mesh, with the desired continuity and smoothness across the patch boundaries. Even just the NURBS-based complex-geometry IGA mesh generation has been a challenge, which has largely been addressed with the Complex-Geometry IGA Mesh Generation (CGIMG) and NURBS Surface-to-Volume Guided Mesh Generation (NSVGMG) methods. The method we are presenting here, Complex-Geometry T-Splines Mesh Generation (CGTSMG), is significantly advancing the state-of-the-art complex-geometry mesh generation from where the CGIMG and NSVGMG brought it. The CGTSMG can, of course, use as input a multipatch NURBS mesh created by the CGIMG or NSVGMG. After the conversion to a T-splines mesh with the desired continuity and smoothness, the mesh quality can further be improved with a good mesh relaxation method like the Fiber-Reinforced Hyperelasticity Mesh Update Method (FRHEMUM). We describe the steps involved in the CGTSMG, followed by the FRHEMUM step, and show that good mesh qualities can be achieved with this mesh generation process.

We present, as a 3D application of recently introduced Complex-Geometry T-Splines Mesh Generation (CGTSMG) method, Space–Time Isogeometric Analysis (ST-IGA) of spacecraft parachute aerodynamics. The computation is for the final design of the Orion spacecraft landing parachute. The parachute canopy has hundreds of gaps and slits, which are modeled, and a wider gap and 16 “windows,” which are resolved. We first generate, manually next to the canopy surfaces and with the Complex-Geometry IGA Mesh Generation elsewhere, a quadratic B-splines mesh made of 670 patches. We then convert that, with the CGTSMG, to a T-splines mesh with C^1 continuity across what used to be the patch boundaries, except around the extraordinary points. After that, we improve the quality by mesh relaxation with the Fiber-Reinforced Hyperelasticity Mesh Update Method. The computation is performed with the ST-IGA and ST Variational Multiscale method. The success of the mesh generation and flow computation process demonstrates the level of sophistication the ST-IGA has reached in complex-geometry flow analysis.

We present the Space–Time Isogeometric Analysis (ST-IGA) of wind turbine rotor and tower aerodynamics, with the rotor geometry of the NREL 5MW offshore baseline wind turbine. The computation is with a given wind speed and a specified rotor speed. The computational challenges include accurate representation of the rotor geometry, multiscale nature of the unsteady flow, the fast, rotational relative motion between the rotor and tower, and the IGA mesh generation for the complex geometry. In addressing the computational challenges, the ST-IGA is used together with the ST Variational Multiscale (ST-VMS) method, which is a core computational method, and the ST Slip Interface (ST-SI) and Complex-Geometry IGA Mesh Generation (CGIMG) methods, which are complementary general-purpose methods. These are the methods of the ST Computational Flow Analysis in this case. The ST-discretization feature provides higher-order accuracy compared to standard discretization methods. The VMS feature addresses the computational challenges associated with the multiscale nature of the unsteady flow. The moving-mesh feature of the ST framework enables high-resolution computation near the blades. The ST-SI enables high-fidelity moving-mesh computations even over meshes made of patches with nonmatching meshes at the interfaces between those patches. The mesh covering the rotor rotates with it, and the SI between the rotating mesh and the rest of the mesh accurately connects the two sides of the solution. The ST-IGA, with IGA basis functions in space, enables more accurate representation of the rotor geometry and increased accuracy in the flow solution. With IGA basis functions in time, it enables more accurate representation of the rotor and mesh rotations. The CGIMG makes it easier in IGA mesh generation to deal with the complex geometry. The computation presented shows that the ST-IGA and the accompanying methods are successful in addressing the challenges and bringing high-fidelity computational analysis to wind turbine rotor and tower aerodynamics.