Tayfun E. Tezduyar

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. [Formula: see text]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.

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 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.

Abstract 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.

Abstract 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.

Abstract 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.

Abstract The Space–Time Computational Flow Analysis (STCFA) started in 1990 with the inception of the Deforming-Spatial-Domain/Stabilized Space–Time (DSD/SST) method. The DSD/SST was introduced as a moving-mesh method for flows with moving boundaries and interfaces, which is a wide class of problems that includes fluid–particle interactions, fluid–structure interactions (FSI), and free-surface and multi-fluid flows. The first 3D computations were reported in 1992. The original DSD/SST method is now called “ST-SUPS,” reflecting its stabilization components. As the STCFA evolved, advanced mesh moving methods, FSI coupling methods, and problem-class-specific methods were introduced to increase its scope and the ST Variational Multiscale was introduced to upgrade its stabilization components to the VMS. Complementary general-purpose methods developed in the evolution of the STCFA include the ST Isogeometric Analysis (ST-IGA) and the ST Slip Interface (ST-SI) and ST Topology Change (ST-TC) methods. The ST-IGA delivers superior accuracy through IGA basis functions not only in space but also in time. 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 ST-TC enables high-fidelity moving-mesh computations even in the presence of topology changes in the fluid mechanics domain, such as an actual contact between moving solid surfaces. The STCFA brought first-of-its-kind solutions in many classes of problems, ranging from fluid–particle interactions in particle-laden flows to FSI in parachute aerodynamics, flapping-wing aerodynamics of an actual locust to ventricle-valve-aorta flow analysis to car and tire aerodynamics with near-actual geometries, road contact, and tire deformation. With the success we see in so many classes of problems, we can conclude that the STCFA has reached a level of remarkable sophistication, scope, and practical value. We present a chronological catalog of the methods and solutions in the STCFA. In Part I of this two-part article, we focus on the methods and solutions in finite element analysis.

Abstract This is Part II of a two-part article that serves as a chronological catalog of the methods and solutions in the Space–Time Computational Flow Analysis (STCFA). In Part I , we focused on the methods and solutions in finite element analysis. Here, we focus on the methods and solutions in isogeometric analysis (IGA). The methods we cover include the ST-IGA and ST Slip Interface method. The first-of-its-kind solutions we cover include the flapping-wing aerodynamics with the wing motion coming from an actual locust, ventricle-valve-aorta flow analysis with patient-specific aorta and realistic ventricle and leaflet geometries and motion, and car and tire aerodynamics with near-actual car body and tire geometries, road contact, and tire deformation. These and the other first-of-its-kind solutions covered show how the STCFA brought solutions in so many classes of challenging flow problems.

Abstract The space–time (ST) computational method “ST-SI-TC-IGA” and recently-introduced complex-geometry isogeometric analysis (IGA) mesh generation methods have enabled high-fidelity computational analysis of tire aerodynamics with near-actual tire geometry, road contact, tire deformation, and aerodynamic influence of the car body. The tire geometries used in the computations so far included the longitudinal and transverse grooves. Here, we bring the tire geometry much closer to an actual tire geometry by using a complex, asymmetric tread pattern. The complexity of the tread pattern required an updated version of the NURBS Surface-to-Volume Guided Mesh Generation (NSVGMG) method, which was introduced recently and is robust even in mesh generation for complex shapes with distorted boundaries. 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). They all play a key role. The ST-TC, uniquely offered by the ST framework, enables moving-mesh computation even with the topology change created by the contact between the tire and the road. It deals with the contact while maintaining high-resolution flow representation near the tire.The computational analysis we present is the first of its kind and shows the effectiveness of the ST-SI-TC-IGA and NSVGMG in tire aerodynamic analysis with complex tread pattern, road contact, and tire deformation.

Variational multiscale methods and their precursors, stabilized methods, which are sometimes supplemented with discontinuity-capturing (DC) methods, have been playing their core-method role in flow computations increasingly with isogeometric discretization. The stabilization and DC parameters embedded in most of these methods play a significant role. The parameters almost always involve some local-length-scale expressions, most of the time in specific directions, such as the direction of the flow or solution gradient. The direction-dependent expressions introduced earlier target B-splines meshes for complex geometries. The key stages of deriving these expressions are mapping the direction vector from the physical element to the parent element in the parametric space, accounting for the discretization spacing along each of the parametric coordinates, and mapping what has been obtained back to the physical element. Here, we extend the local-length-scale calculation method to meshes built from T-splines. T-splines meshes are a superset of B-splines meshes. They provide smooth basis functions in complex geometry and effective refinement without subdividing where we do not need higher resolution. In this article, we focus on the product form T-splines basis functions. They are represented individually in product form, from multiplication of [Formula: see text] 1D basis functions, where [Formula: see text] is the number of parametric dimensions. Each 1D basis function comes from the set of functions associated with one of the parametric directions and the set of functions is defined considering the T-splines nature of the mesh. The product-form basis functions satisfy the partition of unity without using rational functions. For these T-splines, based on the method introduced for B-splines, the local length scales are calculated with Bézier-extraction row operators, which are element-level constants. Using T-splines involves element splitting also for increased integration accuracy. Our local-length-scale expressions are invariant with respect to element splitting performed for integration accuracy but account for the element splitting that is for enhancing the function space.

Abstract The NURBS Surface-to-Volume Guided Mesh Generation (NSVGMG) is a general-purpose mesh generation method, introduced to increase the scope of isogeometric analysis in computing complex-geometry problems. In the NSVGMG, NURBS patch surface meshes serve as guides in generating the patch volume meshes. The interior control points are determined independent of each other, with only a small subset of the surface control points playing a role in determining each interior point. In the updated version of the NSVGMG we are introducing in this article, in the process of determining the location of an interior point in a parametric direction, more weight is given to the closer guides, with the closeness measured along the guides in the other parametric directions. Tests with 2D and 3D shapes show the effectiveness of the NSVGMG in generating good quality meshes, and the robustness of the updated NSVGMG even in mesh generation for complex shapes with distorted boundaries.

Abstract This is Part II of a multipart article on a hyperelastic extended Kirchhoff–Love shell model with out-of-plane normal stress. We introduce an isogeometric discretization method for incompressible materials and present test computations. Accounting for the out-of-plane normal stress distribution in the out-of-plane direction affects the accuracy in calculating the deformed-configuration out-of-plane position, and consequently the nonlinear response of the shell. The return is more than what we get from accounting for the out-of-plane deformation mapping. The traction acting on the shell can be specified on the upper and lower surfaces separately. With that, the model is now free from the “midsurface’ location in terms of specifying the traction. In dealing with incompressible materials, we start with an augmented formulation that includes the pressure as a Lagrange multiplier and then eliminate it by using the geometrical representation of the incompressibility constraint. The resulting model is an extended one, in the Kirchhoff–Love category in the degree-of-freedom count, and encompassing all other extensions in the isogeometric subcategory. We include ordered details as a recipe for making the implementation practical. The implementation has two components that will not be obvious but might be critical in boundary integration. The first one is related to the edge-surface moment created by the Kirchhoff–Love assumption. The second one is related to the pressure/traction integrations over all the surfaces of the finite-thickness geometry. The test computations are for dome-shaped inflation of a flat circular shell, rolling of a rectangular plate, pinching of a cylindrical shell, and uniform hydrostatic pressurization of the pinched cylindrical shell. We compute with neo-Hookean and Mooney–Rivlin material models. To understand the effect of the terms added in the extended model, we compare with models that exclude some of those terms.

Flows with moving boundaries and interfaces (MBI) are a large class of problems that include fluid–particle and fluid–structure interactions, and in broader terms, moving solid surfaces. They also include multi-fluid flows, and as a special case of that, free-surface flows, sometimes in combination with moving solid surfaces. In some classes of MBI problems the solid surfaces could be in fast, linear or rotational relative motion or could come into contact. In almost all real-world applications, the solid surfaces would have complex geometries. All these problems are frequently encountered in engineering analysis and design, pose some of the most formidable computational challenges, and have a common core computational technology need. Bringing solution and analysis to them motivated the development of a good number of core computational methods and special methods targeting specific classes of MBI problems. This paper is an overview of some of those core and special methods. The focus is on isogeometric analysis, complex geometries, incompressible-flow Space–Time Variational Multiscale (ST-VMS) and Arbitrary Lagrangian–Eulerian VMS (ALE-VMS) methods, compressible-flow ST Streamline-Upwind/Petrov–Galerkin (ST-SUPG) and ALE-SUPG methods, and some of the special methods developed in connection with these core ST and ALE methods. The incompressible-flow ST-VMS and ALE-VMS and compressible-flow ST-SUPG and ALE-SUPG are moving-mesh methods, where the mesh moves to have mesh-resolution control near the fluid–solid interfaces, enabling high-resolution boundary-layer representation, an essential feature when the accuracy in representing the boundary layer is a priority. The computational examples presented are car and tire aerodynamics with road contact and tire deformation, ventricle–valve–aorta flow, and gas turbine flow.

Abstract In computation of flow problems with moving solid surfaces, moving-mesh methods such as the space–time (ST) variational multiscale method enable mesh-resolution control near the solid surfaces and thus high-resolution boundary-layer representation. There was, however, a perception that in computations where the solid surfaces come into contact, high-resolution boundary-layer representation and actual-contact representation without leaving a mesh protection opening between the solid surfaces were mutually exclusive objectives in a practical sense. The introduction of the ST topology change (ST-TC) method in 2013 changed the perception. The two objectives were no longer mutually exclusive. The ST-TC makes moving-mesh computation possible even without leaving a mesh protection opening. The contact is represented as an actual contact and the boundary layer is represented with high resolution. Elements collapse or are reborn as needed, and that is attainable in the ST framework while retaining the computational efficiency at a practical level. The ST-TC now has a 10-year history of achieving the two objectives that were long seen as mutually exclusive. With the ST-TC and other ST computational methods introduced before and after, it has been possible to address many of the challenges encountered in conducting flow analysis with boundary layer and contact representation, in the presence of additional intricacies such as geometric complexity, isogeometric discretization, and rotation or deformation of the solid surfaces. The flow analyses conducted with these ST methods include car and tire aerodynamics with road contact and tire deformation and ventricle-valve-aorta flow. To help widen awareness of these methods and what they can do, we provide an overview of the methods, including those formulated in the context of isogeometric analysis, and the computations performed over the 10-year history of the ST-TC.

The stabilization parameters of the methods like the Streamline-Upwind/Petrov–Galerkin, Pressure-Stabilizing/Petrov–Galerkin, and the Variational Multiscale method typically involve two local length scales. They are the advection and diffusion length scales, appearing in the expressions for the advective and diffusive limits of the stabilization parameter. The advection length scale has always been in the flow direction. The diffusion length scales in use have mostly been just element-geometry-dependent, but there is good justification for also making that direction-dependent, so that the spatial variation of the solution is taken into account somehow. The length scale in the solution-gradient direction, which was introduced in 2001, was intended for making sure that near solid surfaces, the element length in the surface-normal direction is selected even if that is not the minimum element length. It was also intended for making sure that in a 2D computation with a 3D mesh, there would be no dependence on the element length in the third direction. With the same objectives, and with better invariance properties, we are now introducing the direction-dependent diffusion length scale calculated from the strain-rate tensor. We accomplish those objectives, get invariance with respect to switching to a different inertial reference frame, and the element length in the surface-normal direction, even when the surface is undergoing rotation, is selected as the diffusion length scale.