This is made possible by the amazing power of CC using only basic tools of calculus combined with computing.We hope the reader will have a good productive time reading the book and also trying out the G2 FEniCS software on old and new challenges. For inspiration a vast material of G2 simulations of turbulent flows is available on the web page of the book at www.bodysoulmath.org.The authors would like to thank the participants of the 2006 Geilo Winter School in Computational Mathematics, who offered valuable comments on the manuscript, and who helped in tracking down some of the mistakes.The first author would like to acknowledge the joint work with Prof. Jonathan Goodman at the Courant Institute in developing the mesh smoothing algorithm of Section 32.5.The main source of mathematicians pictures is the MacTutor History of Mathematics archive, other pictures are taken from what is assumed to be the public domain, or otherwise the sources are stated in the picture captions.

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Abstract. We compute the time average of the drag in two benchmark bluff body problems: a surface mounted cube at Reynolds number 40000, and a square cylinder at Reynolds number 22000, using Adaptive DNS/LES. In Adaptive DNS/LES the Galerkin least-squares finite element method is used, with adaptive mesh refinement until a given stopping criterion is satisfied. Both the mesh refinement criterion and the stopping criterion are based on a posteriori error estimates of a given output of interest, in the form of a space-time integral of a computable residual multiplied by a dual weight, where the dual weight is obtained from solving an associated dual problem computationally, with the data of the dual problem coupling to the output of interest. No filtering is used, and in particular no Reynolds stresses are introduced. We thus circumvent the problem of closure, and instead we estimate the error contribution from subgrid modeling a posteriori, which we find to be small. We are able to predict the mean drag with an estimated tolerance of a few percent using about 10 5 mesh points in space, with the computational power of a PC.

Due to advances in medical imaging, computational fluid dynamics algorithms and high performance computing, computer simulation is developing into an important tool for understanding the relationship between cardiovascular diseases and intraventricular blood flow. The field of cardiac flow simulation is challenging and highly interdisciplinary. We apply a computational framework for automated solutions of partial differential equations using Finite Element Methods where any mathematical description directly can be translated to code. This allows us to develop a cardiac model where specific properties of the heart such as fluid-structure interaction of the aortic valve can be added in a modular way without extensive efforts. In previous work, we simulated the blood flow in the left ventricle of the heart. In this paper, we extend this model by placing prototypes of both a native and a mechanical aortic valve in the outflow region of the left ventricle. Numerical simulation of the blood flow in the vicinity of the valve offers the possibility to improve the treatment of aortic valve diseases as aortic stenosis (narrowing of the valve opening) or regurgitation (leaking) and to optimize the design of prosthetic heart valves in a controlled and specific way. The fluid-structure interaction and contact problem are formulated in a unified continuum model using the conservation laws for mass and momentum and a phase function. The discretization is based on an Arbitrary Lagrangian-Eulerian space-time finite element method with streamline diffusion stabilization, and it is implemented in the open source software Unicorn which shows near optimal scaling up to thousands of cores. Computational results are presented to demonstrate the capability of our framework.

SUMMARYIn this paper, we identify and propose solutions for several issues encountered when designing a mesh adaptation package, such as mesh-to-mesh projections and mesh database design, and we describe an algorithm to integrate a mesh adaptation procedure in a physics solver. The open-source MAdLib package is presented as an example of such a mesh adaptation library. A new technique combining global node repositioning and mesh optimization in order to perform arbitrarily large deformations is also proposed. We then present several test cases to evaluate the performances of the proposed techniques and to show their applicability to fluid-structure interaction problems with arbitrarily large deformations.

We present a framework for adaptive finite element computation of turbulent flow and fluid-structure interaction, with focus on general algorithms that allow for complex geometry and deforming domains. We give basic models and finite element discretization methods, adaptive algorithms and strategies for efficient parallel implementation. To illustrate the capabilities of the computational framework, we show a number of application examples from aerodynamics, aero-acoustics, biomedicine and geophysics. The computational tools are free to download open source as Unicorn, and as a high performance branch of the finite element problem solving environment DOLFIN, both part of the FEniCS project.

We propose a resolution of d'Alembert's Paradox comparing observation of substantial drag/lift in fluids with very small viscosity such as air and water, with the mathematical prediction of zero drag/lift of stationary irrotational solutions of the incompressible inviscid Euler equations, referred to as potential flow. We present analytical and computational evidence that (i) potential flow cannot be observed because it is illposed or unstable to perturbations, (ii) computed viscosity solutions of the Euler equations with slip boundary conditions initiated as potential flow, develop into turbulent solutions which are wellposed with respect to drag/lift and which show substantial drag/lift, in accordance with observations.

is used to compute the drag coefficient c D for the flow past a sphere at Reynolds number Re = 10 4. Using less than 10 5 mesh points, c D is computed to an accuracy of a few percent corresponding to experimental precision, which is at least an order of magnitude cheaper than standard non-adaptive Large Eddy Simulation LES computations in the literature. Adaptive DNS/LES is a General Galerkin G2 method for turbulent flow, where a stabilized Galerkin finite element method is used to compute approximate solutions to the Navier-Stokes equations, with the mesh being adaptively refined until a stopping criterion is reached with respect to the error in a chosen output of interest, in this paper c D. Both the stopping criterion and the mesh refinement strategy are based on a posteriori error estimates, in the form of a space-time integral of residuals times derivatives of the solution of an associated dual problem, linearized at the approximate solution, and with data coupling to the output of interest. There is no filtering of the equations, and thus no Reynolds stresses are introduced which need modelling. The stabilization in the numerical method is acting as a simple turbulence model.