Mathematical Models in Science and Engineering Alfio Quarteroni
athematical modeling aims to de- production. They can explore new solutions in a scribethedifferentaspectsofthereal very short time period, thus allowing the speed of world, their interaction, and their innovation cycles to be increased. This ensures a dynamics through mathematics. It potential advantage to industries, which can save constitutes the third pillar of science time and money in the development and validation Mand engineering, achieving the fulfillment of the phases. We can state therefore that mathematical two more traditional disciplines, which are theo- modeling and scientific computation are gradual- retical analysis and experimentation. Nowadays, ly and relentlessly expanding in manifold fields, mathematical modeling has a key role also in fields becoming a unique tool for qualitative and quanti- such as the environment and industry, while its tative analysis. In the following paragraphs we will potential contribution in many other areas is be- discuss the role of mathematical modeling and of coming more and more evident. One of the reasons scientific computation in applied sciences; their for this growing success is definitely due to the importance in simulating, analyzing, and decision impetuous progress of scientific computation; this making; and their contribution to technological discipline allows the translation of a mathemat- progress. We will show some results and under- ical model—which can be explicitly solved only line the perspectives in different fields such as occasionally—into algorithms that can be treated industry, environment, life sciences, and sports. and solved by ever more powerful computers. See Figure 1 for a synthetic view of the whole process Scientific Computation for Technological leading from a problem to its solution by scientific Innovation computation. Since 1960 numerical analysis—the Linked to the incredible increase of computer discipline that allows mathematical equations (al- calculation speed, scientific computation may be gebraic, functional, differential, and integrals) to decisive enough to define the border between be solved through algorithms—had a leading role complex problems that can be treated and those in solving problems linked to mathematical model- that, on the contrary, cannot. The aim of scien- ing derived from engineering and applied sciences. tific computation is the development of versatile Following this success, new disciplines started to and reliable models, detailed in closed form, and use mathematical modeling, namely information tested on a wide range of test cases, either analog- and communication technology, bioengineering, ical or experimental, for which there are helpful financial engineering, and so on. As a matter of reference solutions. fact, mathematical models offer new possibilities A mathematical model must be able to address to manage the increasing complexity of technol- universal concepts, such as, for instance, the con- ogy, which is at the basis of modern industrial servation of mass or the momentum of a fluid, or Alfio Quarteroni is professor of mathematics at the Ecole the moment of inertia of a structure; moreover, in Polytechnique Fédérale de Lausanne and the Politecni- order to obtain a successful numerical simulation, co of Milan. His email address is alfio.quarteroni@ it is necessary to define which level of detail must epfl.ch. be introduced in the different parts of a model
10 Notices of the AMS Volume 56, Number 1 Figure 1. Scientific computing at a glance. and which simplifications must be carried out contained in the instrumentation on board. Mod- to facilitate its integration into different models. els are used to simulate the stresses and the Models able to simulate very complex problems deformation of some parts of the aircraft (for should take into account uncertainty due to the the simulation of the analysis of materials strain), lack of data (or data affected by noise) that feed through algorithms for the interaction between the model itself. These kinds of models will be fluid and structure with the aim of improving used to foresee natural, biological, and environ- structural and dynamic stability. Similar analyses mental processes, in order to better understand are studied in the car industry, where numerical how complex phenomena work, and also to con- simulation is used in virtually every aspect of tribute to the design of innovative products and design and car production. Models are used to technologies. simulate internal engine combustion in order to An important aspect of scientific computation save fuel, improve the quality of emissions, and is represented by computational fluid dynamics reduce noise. Moreover, to improve performance, (CFD), a discipline that aims to solve by computers security, and comfort, several kinds of equations problems governed by fluids. In aerospace, for must be solved, such as those modeling external example, CFD can be applied in many ways. Nu- and internal fluid dynamics, aero-elasticity, and merical models based on potential flow equations aero-acoustic vibration dynamics, but also those or on the more sophisticated Euler or Navier- governing thermal exchange, combustion process- Stokes equations can be used, for example, in the es, shock waves (occurring during the opening aerodynamic analysis of wing tips or for the whole phase of an air bag), structural dynamics under fuselage for performance optimization. See Figure large stresses, and large deformations to simulate 2 and Figure 3 for numerical simulations carried the consequences of car crashes. The chemical out on, respectively, a civil aircraft (the Falcon 50) industry uses mathematical models to simulate and the X29 experimental aircraft using the Euler polymerization processes, pressing, or extrusion equations solved by a stabilized finite element for complex rheologic materials, where the typi- approximation [1]. Simulation implies validation cal macro analysis of continuum mechanics must and optimization, with the aim of designing air- be connected to the micro one, the latter being craft able to meet certain requirements: better more adequate to describe the complex rheology structural reliability, better aerodynamic perfor- of materials with nanostructure. This requires the mance, lower environmental impact thanks to development of multiscale analysis techniques and the reduction in noise emissions (in the case of algorithms,whichareabletodescribetheexchange commercial airplanes), speed optimization, and of mechanical, thermal, and chemical processes improvement of maneuverability (in the case of in heterogeneous spatial scales. In the electronics military aircraft). The solution to these problems industry, the simulation of drift-diffusion, hydro- requires multi-objective optimization algorithms: dynamics, Boltzmann, or Schrödinger equations deterministic, stochastic, or genetic. Moreover, plays a key role in designing ever smaller and models of electromagnetic diffusion are used to faster integrated circuits, with growing function- simulate external electromagnetic fields in order ality and with dramatic waste reduction (which to restrain them from interfering with those gen- are fundamental, for example, in different appli- erated by the several electronic circuits that are cations of mobile phones). Efficient algorithms are
January 2009 Notices of the AMS 11 Figure 4. Wind velocity simulation over the Mediterranean Sea.
Figure 2. Mach number distribution and either a continental or worldwide scale) there are streamlines for a civil aircraft. changes due to deterministic phenomena, for ex- ample to variation in the inclination of the earth’s axis, the eccentricity of the earth’s orbit, the ocean- ic circulation, or intense geological phenomena like volcanic eruptions. The meteorological prediction problem was formulated as a mathematical issue only at the beginning of the twentieth century by the Norwegian mathematician Vilhelm Bjerkned, who described atmospheric motion using the Euler equations for perfect gas dynamics (well known at that time), suitably modified in order to take into account the action of the force of gravity and the earth’s rotation. Unluckily, data regarding the atmosphere were available only in a few points, and they referred to heterogeneous variables and to different periods of time. Moreover, Euler equations described an ex- tremely wide range of atmospheric motions, which can take place on spatial and temporal scales that Figure 3. Mach number and streamlines on the are very different from each other (feet instead X29 experimental aircraft. of miles, seconds instead of days). The lack of data regarding some of these scales may gen- erate spurious motions (which do not exist in nature) and reduce the prediction quality. A re- useful also for coding and decoding multi-user alistic description of meteorological phenomena messages. cannot but take into account the prediction of water steam distribution, its changes (from liquid Modeling the Weather to gas), and consequent rainfall. The first attempt In the last few decades, the critical problem of to solve this problem from a numerical point of predicting the weather in a short time (daily or view was carried out by Lewis Richardson, who weekly) has become more and more linked to long- succeeded in calculating a concrete example of term prediction (for a decade or even a century), the solution of atmospheric motion on a region to climatic evolution, and to atmospheric pollu- as wide as the whole of North Europe. The re- tion problems. Luckily, there are natural climatic sults obtained by Richardson through extremely changes in a particular area that obey physical law, complicated hand calculations led to completely and can thus be simulated through mathematical wrong predictions, though: as a matter of fact, at models. Also, from a global point of view (over that time there was no theory able to dominate
12 Notices of the AMS Volume 56, Number 1 the traps of the equations to be solved. The contri- bution of Carl-Gustaf Rossby, one of Richardson’s students, was decisive enough to optimize the efforts made by Richardson. After immigrating to the USA in the 1920s, he contributed to founding the meteorological service for civil and military aviation during the Second World War. Among the indirect contributions he gave, the weather prediction made by the Americans for D-Day (June 4, 1944) can be included. The simplified mathe- matical models introduced by Rossby allowed the first meteorological prediction to be made with an electronic computer, resulting from cooperation between John von Neumann and Jules Charney, which started in Princeton in the 1940s. In par- Figure 5. Computed velocity profiles ticular, it was possible to make a prediction for downstream a carotid bifurcation. the whole of North America through a simplified model that described the atmosphere as a unique fluid layer. Even though it took 24 hours to make a prediction for the following 12 hours on the only electronic computer available (ENIAC), the efforts of von Neumann and Charney showed for the first time that a prediction based only on a mathe- matical model could achieve the same results as thosebyanexpertonmeteorologyofthattime.The modern approach to numerical weather prediction was born. As a matter of fact, beyond the spec- tacular improvements in computer performance, there have also been radical improvements in the accuracy of mathematical prediction tools, the development of a theory on the predictability of chaotic dynamical systems, and an improvement in data assimilation techniques. In the 1970s, the systematic use of surveys made by satellites was Figure 6. Shear stress distribution on a introduced, and it constitutes nowadays the most pulmonary artery. relevant part of the data used to start numerical models. Since then, the impact of scientific and technological progress has been very important. For instance, a physical magnitude such as the For instance, the IFS global model of the Euro- shearstress on the endothelial membrane,which is pean Center for Medium Range Weather Forecasts verydifficulttotestinvitro,canbeeasilycalculated (ECMWF) uses a computation grid with an average on real geometries obtained with tri-dimensional spatial resolution of about 22km horizontal and algorithms, thanks to the support of modern 90km vertical. This allows part of the stratosphere and noninvasive data acquisition technology (such to be included. This model can make a 10-day as nuclear magnetic resonance, digital angiogra- prediction in about 1 hour on a modern parallel phy, axial tomography, and Doppler anemometry). supercomputer, even though 6 further hours, nec- Flowing in arteries and veins, blood mechanically essary to insert the data, must be added. The IFS interacts with vessel walls, generating complex model allows reliable predictions to be made for fluid-structural interaction problems. As a matter about 7.5 days on a continental scale in Europe. of fact, the pressure wave transfers mechanical See Figure 4 for an example of weather prediction. energy to the walls, which dilate; such an energy is returned to the blood flow while the vessels Models for Life Sciences are compressed. Vascular simulation of the in- In the 1970s, in vitro experiments, and those teraction between the fluid and the wall requires on animals, represented the main approach to algorithms that describe both the energy transfer cardiovascular studies. Recently, the progress of between the fluid (typically modeled by the Navier- computational fluid dynamics and the great im- Stokes equations) and the structure (modeled by provements of computer performance produced solid mechanics equations) at a macroscopic lev- remarkable advances that revolutionize vascular el, and the influence—at a microscopic level—of research [7]. the shear stress on orientation, deformation, and
January 2009 Notices of the AMS 13 Figure 7. Scientific computing for cardiovascular flow simulation and related topics.
damage of endothelial cells [8]. At the same time, changes in the flow field, generated, for example, flow equations must be coupled to appropriate by the presence of a stenosis, i.e., an artery nar- models in order to describe the transport, diffu- rowing. In the cardiovascular system, conditions sion, and absorption of chemical components in of separated flow and secondary circulatory mo- the blood (such as oxygen, lipids, and drugs), in tions are met, not only in the presence of vessels the different layers that constitute artery walls featuring large curvature (e.g., the aortic bend or (tunica intima, tunica media, and tunica adventi- the coronary arteries), but also downstream of tia). Numerical simulations of this kind may help bifurcations (for instance the carotid artery in its to clarify biochemical modifications produced by internal and external branches) or regions with
14 Notices of the AMS Volume 56, Number 1 restrictions due to the presence of stenosis. There are other areas with a flow inversion (from distal to proximal regions) and also areas with low shear stress with temporal oscillations [9]. These cases are nowadays recognized as potential factors in the development of arterial pathologies. A detailed comprehension of local haemodynamic change, of the effects of vascular wall modifications on the flow scheme, and of the gradual adaptation in the medium to long period of the whole system following surgical interventions, is nowadays pos- sible thanks to the use of sophisticated computer simulations, and may be extremely useful in the preliminary phase before a therapeutic treatment. A similar scenario may provide specific data for surgical procedures. Simulating the flow in a coro- nary bypass, in particular the re-circulation that takes place downstream of the graft in the coro- nary artery, may help us to understand the effects of artery morphology on the flow and thus of Figure 8. Pressure distribution around yacht the post-surgical progression. The theory of op- appendages. timal shape control may be useful for designing a bypass able to minimize the vorticity produced downstream of the graft in the coronary artery. geometric shapes have been standardized, and Similarly, the study of the effects of a vascular even the smallest details may make a difference prosthesis and of implantation of artificial heart from the results point of view. Quoting Jerome valves on local and global haemodynamics may progress thanks to more accurate simulations in Milgram, a professor from MIT and an expert in the field of blood flow. In virtual surgery, the advising different American America’s Cup teams: result of alternative treatments for a specific pa- “America’s Cup teams require an extreme preci- tient may be planned through simulations. This sion in the design of the hull, the keel, and the numerical approach is an aspect of a paradigm of sails. A new boat able to reduce the viscous re- practice, known as predictive medicine. See Fig- sistance by one percent, would have a potential ure 7 for a comprehensive picture on our current advantage on the finish line of as much as 30 research projects in the field of cardiovascular seconds.” To optimize a boat’s performance, it is flow simulations. necessary to solve the fluid-dynamics equations around the whole boat, taking into account the Models for Simulation and Competition variability of wind and waves, of the different The application of mathematical models is not conditions during the yacht race, of the position, limited to the technological, environmental, or and of the moves of the opposing boat, but also medical field. As a matter of fact, deterministic the dynamics of the interaction between fluids and stochasticmodels have beenadoptedfor many (water and air) and the structural components years in analyzing the risk of financial products, (hull, appendages, sails, and mast) must be con- thus facilitating the creation of a new discipline sidered. Moreover, the shape and dynamics of known as financial engineering. Moreover, the new the so-called free surface (the interface between frontier has already begun to touch sociology, air and water) has to be accurately simulated as architecture, free time, and sports. well. A complete mathematical model must take As far as competitive sports are concerned, CFD into account all these aspects characterizing the for some years now has been assuming a key role physical problem. The aim is to develop together in analyzing and designing Formula One cars. But with the designers optimal models for the hull, Formula One racing is not the only field where the keel, and appendages. mathematical/numerical modelling has been ap- Ideally, one wishes to minimize water resistance plied. As a matter of fact, my research group from on the hull and appendages and to maximize the EPFL has been involved in an extremely interesting boost produced by the sails. Mathematics allows experience, which saw the Swiss yacht Alinghi win different situations to be simulated, thus reducing the America’s Cup both in 2003 and again in 2007. costs and saving time usually necessary to pro- Until twenty years ago, the different designing duce a great number of prototypes to be tested in teams used to develop different shapes of sails, a towing tank and wind tunnel. For each new boat hulls, bulbs, and keels. Nowadays the different simulation proposed by the designers (which were
January 2009 Notices of the AMS 15 turbulence vorticity generated by the interaction of the air, thus obtaining useful information for the tactician as well. These studies aim to design a boat having an optimal combination of the four features that an America’s Cup yacht must have: lightness, speed, resistance, and maneuverability necessary to change the race outcome. A more in-depth description of the mathemat- ical tools necessary for this kind of investigation is provided in the next section.
Mathematical Models for America’s Cup The standard approach adopted in the America’s Cup design teams to evaluate whether a design change (and all the other design modifications that this change implies) is globally advantageous, is based on the use of a Velocity Prediction Program (VPP), which can be used to estimate the boat speed and attitude for any prescribed wind con- Figure 9. Streamlines around mainsails and dition and sailing angle. A numerical prediction spinnaker in a downwind leg. of boat speed and attitude can be obtained by modeling the balance between the aerodynamic and hydrodynamic forces acting on the boat. several hundred), it was necessary to build the ge- For example, on the water plane, a steady sailing ometrical model—about 300 splines surfaces are condition is obtained imposing two force balances needed to overlay the whole boat—to create the in the x direction (aligned with the boat velocity) grid on the surface of all the elements of the boat and the y direction (normal to x on the water reliable enough to enable the determination of plane) and a heeling moment balance around the the transition between laminar flow and turbulent centerline of the boat: flow regions, and consequently to generate the volumetric grid in external domain. The Navier- Dh + T a = 0,
Stokes equations for incompressible viscous flows (1) Sh + Sa = 0, must be used to describe both water and wind Mh + Ma = 0, dynamics and the consequent free surface, which need to be completed by additional equations that where Dh is the hydrodynamic drag (along the allow the computation of turbulent energy and course direction), T a is the aerodynamic thrust, its dissipation rate. These equations cannot be Sh is the hydrodynamic side force perpendicular solved exactly to yield explicit solutions in closed to the course, Sa is the aerodynamic side force, form. Their approximate solution requires the Mh and M a are, respectively, the hydromechani- introduction of refined discretization methods, cal righting moment and the aerodynamic heeling which allow an infinite dimensional problem to moment around the boat mean line. The angle be transformed into a big but finite dimensional βY between the course direction and the boat one. The typical calculation, based on finite vol- centerline is called yaw angle. The aerodynamic ume schemes, involved the solution of nonlinear thrust and side force can be seen as a decom- problems with many millions of unknowns. Using position in the reference system aligned with the parallel algorithms, 24 hours on parallel calcula- course direction of the aerodynamic lift and drag, tion platforms with 64 processors were necessary which are defined on a reference system aligned to produce a simulation, characterized by more with the apparent wind direction. Similar balance than 160 million unknowns. A further compu- equations can be obtained for the other degrees tation is concerned with the simulation of the of freedom. dynamical interaction between wind and sails by In a VPP program, all the terms in system fluid-structure algorithms. These simulations en- (1) are modeled as functions of boat speed, heel able the design team to eliminate those solutions angle, and yaw angle. Suitable correlations be- that seem innovative and to go on with those that tween the degrees of freedom of the system and actually guarantee better performance. Moreover, the different force components can be obtained by simulating the effects of aerodynamic interac- based on different sources of data: experimen- tion between two boats, one can determine the tal results, theoretical predictions, and numerical consistency of shadow regions (the areas with less simulations. wind because of the position of a boat with re- The role of advanced computational fluid dy- spect to the other), the flow perturbation, and the namics is to supply accurate estimates of the
16 Notices of the AMS Volume 56, Number 1 Aero Total Force
Aero Side Force Heeling Moment Aero Lift Aero Side Force
Aero Drag Aero Thrust Righting Boat Speed Hydro Side Force Hydro Drag Moment
True Wind z Apparent Wind Weight Buoyancy Force Y Hydro Side Force Y X Hydro Total Force
Figure 10. Forces and moments acting on boat.
forces acting on the boat in different sailing con- w ) and ρa (in a). The values of ρw and ρa depend ditions in order to improve the reliability of the on the fluid temperatures, which are considered prediction of the overall performance associated toΩ be constantΩ in the present model. The fluid with a given design configuration. viscosities µw (in w ) and µa (in a) are constants that depend on ρw and ρa, respectively. The flow equations The set of equationsΩ (2)-(4)Ω can therefore be Let denote the three-dimensional computational seen as a model for the evolution of a two-phase domain in which we solve the flow equations. If ˆ flow consisting of two immiscible incompressible Ω is a region surrounding the boat B, the computa- fluids with constant densities ρw and ρa and dif- Ω tional domain is the complement of B with respect ferent viscosity coefficients µw and µa. In this to ˆ, that is = ˆ\B. The equations that gov- respect, in view of the numerical simulation, we ern the flow around B are the density-dependent could regard equation (2) as the candidate for (orΩ inhomogeneous)Ω Ω incompressible Navier–Stokes updating the (unknown) interface location , then equations, which read: treat equations (3)-(4) as a coupled system of Navier–Stokes equations in the two sub-domainsΓ and : ∂ρ w a (2) + ∇ · (ρu) = 0 ∂t Ω∂(ρw uw )Ω + ∇ · (ρw uw ⊗ uw ) − ∇ · τw (uw , pw ) = ρw g, ∂(ρu) ∂t (3) + ∇ · (ρu ⊗ u) − ∇ · τ(u, p) = ρg ∂t ∇ · uw = 0,
(4) ∇ · u = 0 in w × (0, T ), for x ∈ and 0
(6) τa(ua, pa) · n = τw (uw , pw ) · n + κσ n on , the aerodynamic pressure field deforms the sail surfaces and this, in its turn, modifies the flow where σ is the surface tension coefficient, that is Γ field around the sails. a force per unit length of a free surface element From a mathematical viewpoint, this yields a acting tangentially to the free-surface. It is a prop- coupled system that comprises the incompress- erty of the liquid and depends on the temperature ible Navier-Stokes equations with constant density as well as on other factors. The quantity κ in (6) is ρ = ρ (3-4) and a second order elastodynamic the curvature of the free-surface, κ = R−1 + R−1, air t1 t2 equation that models the sail deformation as that where R and R are radii of curvature along the t1 t2 of a membrane. More specifically, the evolution coordinates (t ,t ) of the plane tangential to the 1 2 of the considered elastic structure is governed free-surface (orthogonal to n). by the classical conservation laws for continuum mechanics. Coupling with a 6-DOF rigid body dynamical Considering a Lagrangian framework, if ˆ is system s the reference 2D domain occupied by the sails, the Ω The attitude of the boat advancing in calm water governing equation can be written as follows: or wavy sea is strictly correlated with its perfor- 2 mance. For this reason, a state-of-the-art numerical ∂ d ˆ (9) ρs = ∇· σs (d) + f s in s × (0,T], tool for yacht design predictions should be able ∂t2
to account for the boat motion. where ρs is the material density, theΩ displacement Following the approach adopted in [2, 3], two d is a function of the space coordinates x ∈ ˆ s and orthogonal cartesian reference systems are con- of the time t ∈ [0; T ], σs are the internal stresses Ω sidered: an inertial reference system (O,X,Y,Z), while f s are the external loads acting on the sails which moves forward with the mean boat speed, (these are indeed the normal stresses τ(u, p)·n on and a body-fixed reference system (G, x, y, z), the sail surface exerted by the flowfield). In fact,
whose origin is the boat center of mass G, which ˆ s represents a wider (bounded and disconnected) translates and rotates with the boat. The XY plane domain that includes also the mast and the yarns Ω in the inertial reference system is parallel to the as parts of the structural model. The boundary of undisturbed water surface, and the Z-axis points + ˆ s is denoted by ∂ ˆ s and [0; T ] ⊂ R is the time upward. The body-fixed x-axis is directed from interval of our analysis. For suitable initial and Ω Ω bow to stern, y positive starboard, and z upwards. boundary conditions and an assignment of an ap- The dynamics of the boat in the 6 degrees of propriate constitutive equation for the considered freedom are determined by integrating the equa- materials (defining σs (d)), the displacement field tions of variation of linear and angular momentum d is computed by solving (9) in its weak form: in the inertial reference system, as follows ∂2d ¨ ρ i (δd )dx + σ II (δǫ )dx (7) mXG = F s 2 i ik ki Zˆs ∂t Zˆ s ¯¯¯−1 ¯¯¯ −1 (10) (8) T¯¯IT¯ ˙ + × T¯¯IT¯ = MG Ω Ω = fs i (δdi )dx, Zˆs where m is the boatΩ Ω mass, X¨ G isΩ the linear accel- eration of the center of mass, F is the force acting where σ II is the secondΩ Piola-Kirchoff stress ten- on the boat, ˙ and are the angular acceleration sor, ǫ is the Green-Lagrange strain tensor, and δd and velocity, respectively, MG is the moment with are the test functions expressing the virtual defor- Ω Ω respect to G acting on the boat, ¯I is the tensor of mation. The second coupling condition enforces ∂d inertia of the boat about the body-fixed reference the continuity of the two velocity fields, u and ∂t , system axes, and T¯ is the transformation matrix on the sail surface. between the body-fixed and the inertial reference system (see [2] for details). Fluid-structural coupling algorithm The forces and moments acting on the boat are As previously introduced, the coupling procedure given by iteratively loops between the fluid solver (passing sail velocities and getting pressure fields) and the F = F Flow + mg + F Ext structural solver (passing pressures and getting = + − × MG M Flow (XExt XG) F Ext velocities and structural deformations) until the
where F Flow and MFlow are the force and moment, structure undergoes no more deformations be- respectively, due to the interaction with the flow cause a perfect balance of forces and moments is and F Ext is an external forcing term (which may reached. When dealing with transient simulations, model, e.g., the wind force on sails) while XExt is this must be true for each time step, and the its application point. sail geometry evolves over time as a sequence of
18 Notices of the AMS Volume 56, Number 1 converged states. On the other hand, a steady sim- [3] , RANSE simulations for sailing yachts in- ulation can be thought of as a transient one with cluding dynamic sinkage & trim and unsteady an infinite time step, such that “steady” means motions in waves, in High Performance Yacht Design a sort of average of the true (unsteady) solution Conference, pages 13–20, Auckland, 2002. over time. More formally, we can define two oper- [4] D. Detomi, N. Parolini, and A. Quarteroni, Math- ematics in the wind, 2008, to appear in Monographs ators called Fluid and Struct that represent the of the Real Academia de Ciencias de Zaragoza, Spain. fluid and structural solvers, respectively. In par- [5] N. Parolini and A. Quarteroni, Mathematical mod- ticular, Fluid can be any procedure that can solve els and numerical simulations for the America’s the incompressible Navier-Stokes equations while Cup, Comp. Meth. Appl. Mech. Eng., 173 (2005), Struct should solve a membrane-like problem, 1001–1026. possibly embedding suitable nonlinear models to [6] Modelling and numerical simulation for yacht take into account complex phenomena such as, for engineering, in Proceedings of the 26th Symposium example, the structural reactions due to a fabric on Naval Hydrodynamics, Arlington, VA, USA, 2007, wrinkle. Strategic Analysis, Inc. The fixed-point problem can be reformulated [7] A. Quarteroni, Modeling the cardiovascular system: A mathematical adventure, SIAM News 34 (2001). with the new operators as follows: [8] A. Quarteroni and L. Formaggia, Mathematical (11) Fluid (Struct(p)) = p. modelling and numerical simulation of the car- diovascular system, Modelling of Living Systems, A resolving algorithm can be devised as follows. Handbook of Numerical Analysis Series, (P. G. Ciarlet At a given iteration the pressure field on sails p and J. L. Lions, eds.), Elsevier, Amsterdam, 2004. is passed to the structural solver (Struct), which [9] A. Quarteroni, L. Formaggia, and A. Veneziani, returns the new sail geometries and the new sail Complex Systems in Biomedicine, Springer, Milano, velocity fields. Afterwards, these quantities are 2006. passed to the fluid solver (Fluid) which returns the same pressure field p on sails. Clearly, the Acknowledgments “equal” sign holds only at convergence. The re- The contributions from Davide Detomi and Nicola sulting fixed-point iteration can be rewritten more Parolini to the section on Mathematical Models for explicitly as follows: Given a pressure field on sails America’s Cup are gratefully acknowledged. p , do: k Credit for graphic images is attributed to: (Gk+1, Uk+1) = Struct(pk) Marzio Sala (Figures 2 and 3) (12) p¯k+1 = Fluid(Gk+1, Uk+1) ARPA Emilia Romagna (Figure 4) pk+1 = (1 − θk)pk + θkp¯k+1 Martin Prosi (Figure 5) Johnathan Wynne (Figure 6) where Gk+1 and Uk+1 are the sail geometry and the sail velocity field at step k + 1, respectively, while Nicola Parolini (Figure 8) Davide Detomi (Figure 9) θk is a suitable acceleration parameter. Even though the final goal is to run an unsteady simulation, the fluid-structure procedure has to run some preliminary steady couplings to provide a suitable initial condition. The steady run iterates until a converged sail shape and flow field are obtained, where converged means that there does exist a value of kc such that (11) is satisfied for every k > kc (within given tolerances on forces and/or displacements). When running steady sim- ulations the velocity of the sails is required to be null at each coupling, thus somehow enforcing the convergence condition (which prescribes null velocities at convergence). This explains why con- vergence is slightly faster when running steady simulations with respect to transient ones (clearly only when such a solution reflects a steady state physical solution).
References [1] A. Quarteroni and A. Valli, Domain Decomposition Methods for Partial Differential Equations, Oxford University Press, Oxford, 1999. [2] R. Azcueta, Computation of turbulent free-surface flows around ships and floating bodies, Ph.d. thesis, 2001.
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