In computational flow analysis with moving solid surfaces and contact between the solid surfaces, it is a challenge to represent the boundary layers with an accuracy attributed to moving-mesh methods and to represent the contact without leaving a mesh protection gap. The space-time topology change (ST-TC) method, introduced in 2013, makes moving-mesh computation possible even when we have contact between moving solid surfaces or other kinds of flow-domain TC. The contact is represented without giving up on high-resolution flow representation near the moving surfaces. 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 this class of flow analysis in the presence of additional complexities such as geometric complexity, rotation or deformation of the solid surfaces and the multiscale nature of the flow. In this first part of a two-part article, we provide an overview of the methods that made all that possible. We also provide an overview of the computations performed for tire aerodynamics with challenges that include the complexity of a near-actual tire geometry with grooves, road contact, tire deformation and rotation, road roughness and fluid films.

In computation of flow problems with moving boundaries and interfaces, including fluid-structure interaction, moving the fluid mechanics mesh to follow the fluid-solid interface enables mesh-resolution control near the interface. Therefore moving-mesh methods, such as the Space-Time Variational Multiscale (ST-VMS) method, enable high-resolution boundary-layer representation near fluid-solid interfaces and thus higher accuracy in such critical flow regions. In flow problems with contact between solid surfaces, until recently, one had to either give up on representing the actual contact and leave a small gap or give up on using a moving-mesh method and thus give up on having high-fidelity flow solution near the solid surfaces. The ST Topology Change (ST-TC) method changed all that. Now we can both represent the actual contact and have high-fidelity flow solution near the solid surfaces. With the ST-VMS, which serves as the core method, and the ST-TC and two other special methods, the ST Slip Interface method and ST Isogeometric Analysis, we have created a powerful computational framework. The new framework is enabling high-fidelity computational flow analysis of some of the most complex problems, such as the ventricle-valve-aorta sequence. This chapter is a description and demonstration of that framework.

We address the computational challenges of and presents results from ventricle-valve-aorta flow analysis. Including the left ventricle (LV) in the model makes the flow into the valve, and consequently the flow into the aorta, anatomically more realistic. The challenges include accurate representation of the boundary layers near moving solid surfaces even when the valve leaflets come into contact, computation with high geometric complexity, anatomically realistic representation of the LV motion, and flow stability at the inflow boundary, which has a traction condition. The challenges are mainly addressed with a Space–Time (ST) method that integrates three special ST methods around the core, ST Variational Multiscale (ST-VMS) method. The three special methods are the ST Slip Interface (ST-SI) and ST Topology Change (ST-TC) methods and ST Isogeometric Analysis (ST-IGA). The ST-discretization feature of the integrated method, ST-SI-TC-IGA, 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 in the LV, valve and aorta. The moving-mesh feature of the ST framework enables high-resolution computation near the leaflets. The ST-TC enables moving-mesh computation even with the TC created by the contact between the leaflets, dealing with the contact while maintaining high-resolution representation near the leaflets. The ST-IGA provides smoother representation of the LV, valve and aorta surfaces and increased accuracy in the flow solution. The ST-SI connects the separately generated LV, valve and aorta NURBS meshes, enabling easier mesh generation, connects the mesh zones containing the leaflets, enabling a more effective mesh moving, helps the ST-TC deal with leaflet–leaflet contact location change and contact sliding, and helps the ST-TC and ST-IGA keep the element density in the narrow spaces near the contact areas at a reasonable level. The ST-SI-TC-IGA is supplemented with two other special methods in this article. A structural mechanics computation method generates the LV motion from the CT scans of the LV and anatomically realistic values for the LV volume ratio. The Constrained-Flow-Profile (CFP) Traction provides flow stability at the inflow boundary. Test computation with the CFP Traction shows its effectiveness as an inflow stabilization method, and computation with the LV-valve-aorta model shows the effectiveness of the ST-SI-TC-IGA and the two supplemental methods.

Heart valve fluid–structure interaction (FSI) analysis is one of the computationally challenging cases in cardiovascular fluid mechanics. The challenges include unsteady flow through a complex geometry, solid surfaces with large motion, and contact between the valve leaflets. We introduce here an isogeometric sequentially-coupled FSI (SCFSI) method that can address the challenges with an outcome of high-fidelity flow solutions. The SCFSI analysis enables dealing with the fluid and structure parts individually at different steps of the solutions sequence, and also enables using different methods or different mesh resolution levels at different steps. In the isogeometric SCFSI analysis here, the first step is a previously computed (fully) coupled Immersogeometric Analysis FSI of the heart valve with a reasonable flow solution. With the valve leaflet and arterial surface motion coming from that, we perform a new, higher-fidelity fluid mechanics computation with the space–time topology change method and isogeometric discretization. Both the immersogeometric and space–time methods are variational multiscale methods. The computation presented for a bioprosthetic heart valve demonstrates the power of the method introduced.

We present detailed studies on mesh refinement influence and cardiac-cycle flow periodicity in aorta flow analysis with isogeometric discretization. Both factors play a key role in the reliability and practical value of aorta flow analysis. The core computational method is the space–time Variational Multiscale (ST-VMS) method. The other key method is the ST Isogeometric Analysis (ST-IGA). The ST framework, in a general context, provides higher-order accuracy. The VMS feature of the ST-VMS addresses the computational challenges associated with the multiscale nature of the unsteady flow in the aorta. The ST-IGA provides smoother representation of the aorta and increased accuracy in the flow solution. We conduct the studies for a patient-specific aorta geometry. We determine the level of mesh refinement needed and assess the nature of the flow periodicity reached.

We present a heart valve flow computation with the Space-Time Slip Interface Topology Change (ST-SI-TC) method and Isogeometric Analysis (IGA). The computation is for a realistic heart valve model with actual contact between the valve leaflets. The ST-SI-TC method integrates the ST-SI and ST-TC methods in the framework of the ST Variational Multiscale (ST-VMS) method. The STVMS method functions as a moving-mesh method, which maintains high-resolution boundary layer representation near the solid surfaces. The ST-TC method was introduced for moving-mesh computation of flow problems with TC, such as contact between the leaflets of a heart valve. It deals with the contact while maintaining highresolution representation near the leaflet surfaces. The ST-SI method was originally introduced to addresses the challenge involved in high-resolution representation of the boundary layers near spinning solid surfaces. The mesh covering a spinning solid surface spins with it, and the SI between that mesh and the rest of the mesh accurately connects the two sides. This maintains the high-resolution representation near solid surfaces. In the context of heart valves, the SI connects the sectors of meshes containing the leaflets, enabling a more effective mesh moving. In that context, the ST-SI-TC method enables high-resolution representation even when the contact is between leaflets that are covered by meshes with SI. It also enables dealing with contact location change or contact and sliding on the SI. With IGA, in addition to having a more accurate representation of the surfaces and increased accuracy in the flow solution, the element density in the narrow spaces near the contact areas is kept at a reasonable level. Furthermore, because the flow representation in the contact area has a wider support in IGA, the flow computation method becomes more robust. The computation we present for an aortic-valve model shows the effectiveness of the ST-SI-TC-IGA method.

We present our computational methods for and results from aorta flow analysis and heart valve flow and structure analysis. In flow analysis, the core method is the space–time Variational Multiscale (ST-VMS) method. The other key methods are the ST Slip Interface (ST-SI) and ST Topology Change (ST-TC) methods and the ST Isogeometric Analysis (ST-IGA). The ST framework, in a general context, provides higher-order accuracy. The VMS feature of the ST-VMS addresses the computational challenges associated with the multiscale nature of the unsteady flows in the aorta and heart valve. The moving-mesh feature of the ST framework enables high-resolution computation near the valve leaflets. The ST-SI connects the sectors of meshes containing the leaflets, enabling a more effective mesh moving. The ST-TC enables moving-mesh computation even with the TC created by the contact between the leaflets. It deals with the contact while maintaining high-resolution representation near the leaflets. Integration of the ST-SI and ST-TC enables high-resolution representation even though parts of the SI are coinciding with the leaflet surfaces. It also enables dealing with leaflet–leaflet contact location change and contact sliding. The ST-IGA provides smoother representation of aorta and valve surfaces and increased accuracy in the flow solution. With the integration of the ST-IGA with the ST-SI and ST-TC, the element density in the narrow spaces near the contact areas is kept at a reasonable level. In structure analysis, we use a Kirchhoff–Love shell model, where we take the stretch in the third direction into account in calculating the curvature term. The computations presented demonstrate the scope and effectiveness of the methods.

Heart valve flow computation requires accurate representation of boundary layers near moving solid surfaces, including the valve leaflet surfaces, even when the leaflets come into contact. It also requires dealing with a high level of geometric complexity. We address these computational challenges with a Space–Time (ST) method developed by integrating three special ST methods in the framework of the ST Variational Multiscale (ST-VMS) method. The special methods are the ST Slip Interface (ST-SI) and ST Topology Change (ST-TC) methods and ST Isogeometric Analysis (ST-IGA). The computations are for a realistic aortic-valve model with prescribed valve leaflet motion and actual contact between the leaflets. The ST-VMS method functions as a moving-mesh method, which maintains high-resolution boundary layer representation near the solid surfaces, including leaflet surfaces. The ST-TC method was introduced for moving-mesh computation of flow problems with TC, such as contact between the leaflets of a heart valve. It deals with the contact while maintaining high-resolution representation near the leaflet surfaces. The ST-SI method was originally introduced to have high-resolution representation of the boundary layers near spinning solid surfaces. The mesh covering a spinning solid surface spins with it, and the SI between the spinning mesh and the rest of the mesh accurately connects the two sides. In the context of heart valves, the SI connects the sectors of meshes containing the leaflets, enabling a more effective mesh moving. In that context, integration of the ST-SI and ST-TC methods enables high-resolution representation even when the contact is between leaflets that are covered by meshes with SI. It also enables dealing with contact location change or contact and sliding on the SI. By integrating the ST-IGA with the ST-SI and ST-TC methods, in addition to having a more accurate representation of the surfaces and increased accuracy in the flow solution, the element density in the narrow spaces near the contact areas is kept at a reasonable level. Furthermore, because the flow representation in the contact area has a wider support in IGA, the flow computation method becomes more robust. The computations we present for an aortic-valve model with two different modes of prescribed leaflet motion show the effectiveness of the ST-SI-TC-IGA method.

We focus on turbocharger computational flow analysis with a method that possesses higher accuracy in spatial and temporal representations. In the method we have developed for this purpose, we use a combination of (i) the Space–Time Variational Multiscale (ST-VMS) method, which is a stabilized formulation that also serves as a turbulence model, (ii) the ST Slip Interface (ST-SI) method, which maintains high-resolution representation of the boundary layers near spinning solid surfaces by allowing in a consistent fashion slip at the interface between the mesh covering a spinning surface and the mesh covering the rest of the domain, and (iii) the Isogeometric Analysis (IGA), where we use NURBS basis functions in space and time. The basis functions are spatially higher-order in all representations, and temporally higher-order in representation of the solid-surface and mesh motions. The ST nature of the method gives us higher-order accuracy in the flow solver, and when combined with temporally higher-order basis functions, a more accurate representation of the surface motion, and a mesh motion consistent with that. The spatially higher-order basis functions give us again higher-order accuracy in the flow solver, a more accurate, in some parts exact, representation of the surface geometry, and better representation in evaluating the second-order spatial derivatives. Using NURBS basis functions with a complex geometry is not trivial, however, once we generate the mesh, the computational efficiency is substantially increased. We focus on the turbine part of a turbocharger, but our method can also be applied to the compressor part and thus can be extended to the full turbocharger.

We present a method for structural and fluid mechanics computations of ram-air parachutes. A ram-air parachute is a parafoil inflated by the airflow through the inlets at the leading edge. It has better control and gliding capability than round parachutes. Reliable analysis of ram-air parachutes requires accurate representation of the parafoil geometry, fabric porosity and the complex, multiscale flow behavior involved in this class of problems. The key components of the method are (i) the Space–Time Variational Multiscale (ST-VMS) method, (ii) the version of the ST Slip Interface (ST-SI) method where the SI is between a thin porous structure and the fluid on its two sides, (iii) the ST Isogeometric Analysis (ST-IGA), and (iv) special-purpose NURBS mesh generation techniques for the parachute structure and the flow field inside and outside the parafoil. The ST-VMS method is a stabilized formulation that also serves as a turbulence model and can deal effectively with the complex, multiscale flow behavior. With the ST-SI version for porosity modeling, we deal with the fabric porosity in a fashion consistent with how we deal with the standard SIs and how we enforce the Dirichlet boundary conditions weakly. The ST-IGA, with NURBS basis functions in space, gives us, with relatively few number of unknowns, accurate representation of the parafoil geometry and increased accuracy in the flow solution. The special-purpose mesh generation techniques enable NURBS representation of the structure and fluid domains with significant geometric complexity. The test computations we present are for building a starting parachute shape and a starting flow field associated with that parachute shape, which are the first two key steps in fluid–structure interaction analysis. The computations demonstrate the effectiveness of the method in this class of problems.