This book is a tutorial written by researchers and developers behind the FEniCS Project and explores an advanced, expressive approach to the development of mathematical software. The presentation spans mathematical background, software design and the use of FEniCS in applications. Theoretical aspects are complemented with computer code which is available as free/open source software. The book begins with a special introductory tutorial for beginners. Followingare chapters in Part I addressing fundamental aspects of the approach to automating the creation of finite element solvers. Chapters in Part II address the design and implementation of the FEnicS software. Chapters in Part III present the application of FEniCS to a wide range of applications, including fluid flow, solid mechanics, electromagnetics and geophysics.

Automated Solution of Differential Equations by the Finite Element Method: The FEniCS Book

The work is giving estimations of the discrepancy between solutions of the initial and the homogenized problems for a one{dimensional second order elliptic operators with random coeecients satisfying strong or uniform mixing conditions. We obtain several sharp estimates in terms of the corresponding mixing coeecient. Abstract. In the theory of homogenisation it is of particular interest to determine the classes of problems which are stable on taking the homogenisation limits. A notable situation where the limit enlarges the class of original problems is known as memory (nonlocal) eeects. A number of results in that direction has been obtained for linear problems. Tartar (1990) innitiated the study of the eeective equation corresponding to nonlinear equation: @ t u n + a n u 2 n = f: Signiicant progress has been hampered by the complexity of required computations needed in order to obtain the terms in power{series expansion. We propose a method which overcomes that diiculty by introducing graphs representing the domain of integration of the integrals in each term. The graphs are relatively simple, it is easy to calculate with them and they give us a clear image of the form of each term. The method allows us to discuss the form of the eeective equation and the convergence of power{series expansions. The feasibility of our method for other types of nonlinearities will be discussed as well.

Applied Mathematics and Computation

In December 2004, a huge tsunami in SE Asia claimed many lives and caused catastrophic damage. This event has stimulated a debate about the role played by coastal ecosystems such as mangrove forests and coral reefs in protecting low-lying coastal areas. While some observers claim that these ecosystems play an important role, others are more sceptical and fear that financial resources may be diverted away from tsunami preparedness programmes to ecosystem rehabilitation schemes. Here, we review the role of coastal ecosystems in mitigating sea wave hazards. In particular, we examine the influence of coastal vegetation in severely affected parts of Aceh and Southern Thailand during the 2004 tsunami, and include direct observations made during two field visits in 2006. Numerous scientific studies illustrate that coastal ecosystems are important in protecting the coast against normal and extreme wind-driven waves, including cyclonic storm surges. However, much less information is available concerning the role of coastal ecosystems in protecting against tsunamis, and most studies have been too limited in scope to be able to generalise over large geographic areas and diverse conditions. In particular, few studies have taken account of the high variability in the energy and speed of tsunami waves along different coastline stretches, making it difficult to compare the role of ecosystems in different regions. Other aspects that have been little studied include the spatial patterns of impact, e.g. flow diversion and channelling as a result of hydraulic resistance, and the effects of tree breaking and flow debris. However, despite these limitations, some tentative conclusions can be drawn about the protection provided by ecosystems against tsunami waves: (1) The most important predictor variables of the tsunami hazard (i.e. set-up and inundation) at the regional and landscape levels are distance to the tsunami source and coastal topography, in particular near-shore bathymetry. (2) The influence of coral reefs on tsunami waves is complex. While closed, intact reefs can provide some protection, water may be accelerated through channels in fragmented reefs, causing even greater destruction on land. (3) In contrast, seagrass beds appear to provide a more consistent buffering against tsunamis, though more data are needed to quantify this effect. (4) In several locations (particularly farther away from the tsunami source), mangroves and other vegetation probably provided some protection against the 2004 tsunami; in numerous other locations, however, vegetation provided no protection or may in some cases even have increased the hazard (e.g. by contributing to flow debris, or by channelling water flows). It seems that whether or not vegetation does provide protection depends on many factors, including stand size, density, species composition, structure and homogeneity. Overall, the value of vegetation as a potential tsunami buffer is probably fairly minor, though it cannot be completely dismissed. Many more detailed spatial and hydro-dynamic analyses are needed before the hazard-vegetation interactions can be realistically modelled and visualised, e.g. on risk maps. (5) Tsunami greenbelts should not be treated as alternatives to early warning systems. Greenbelts may only be considered as an economical (and multi-functional) means to provide relative hazard protection for material assets (e.g. infrastructure, agriculture). We recommend a more holistic view, with tsunami hazard mitigation being seen as one of several services provided by coastal ecosystems. Equally, the value of other ecosystem services may be seen within the frame of risk management as they foster social stability and, in case of disaster, robustness. To assess the overall value of these services and to develop more accurate indicators of risk will require much more transdisciplinary research.

The 2004 tsunami in Aceh and Southern Thailand: A review on coastal ecosystems, wave hazards and vulnerability

A set of model equations for water–wave propagation is derived by piecewise integration of the primitive equations of motion through two arbitrary layers. Within each layer, an independent velocity profile is derived. With two separate velocity profiles, matched at the interface of the two layers, the resulting set of equations has three free parameters, allowing for an optimization with known analytical properties of water waves. The optimized model equations show good linear wave characteristics up to kh ≈ 6 , while the second–order nonlinear behaviour is captured to kh ≈ 6 as well. A numerical algorithm for solving the model equations is developed and tested against one– and two–horizontal–dimension cases. Agreement with laboratory data is excellent.

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A two-layer approach to wave modelling

On the 30th of December 2002 two tsunamis were generated only 7 min apart in Stromboli, southern Tyrrhenian Sea, Italy. They represented the peak of a volcanic crisis that started 2 days before with a large emission of lava flows from a lateral vent that opened some hundreds of meters below the summit craters. Both tsunamis were produced by landslides that detached from the Sciara del Fuoco. This is a morphological scar and is the result of the last collapse of the northwestern flank of the volcanic edifice, that occurred less than 5 ka b.p. The first tsunami was due to a submarine mass movement that started very close to the coastline and that involved about 20×106 m3 of material. The second tsunami was engendered by a subaerial landslide that detached at about 500 m above sea level and that involved a volume estimated at 4–9×106 m3. The latter landslide can be seen as the retrogressive continuation of the first failure. The tsunamis were not perceived as distinct events by most people. They attacked all the coasts of Stromboli within a few minutes and arrived at the neighbouring island of Panarea, 20 km SSW of Stromboli, in less than 5 min. The tsunamis caused severe damage at Stromboli. In this work, the two tsunamis are studied by means of numerical simulations that use two distinct models, one for the landslides and one for the water waves. The motion of the sliding bodies is computed by means of a Lagrangian approach that partitions the mass into a set of blocks: we use both one-dimensional and two-dimensional schemes. The landslide model calculates the instantaneous rate of the vertical displacement of the sea surface caused by the motion of the underwater slide. This is included in the governing equations of the tsunami, which are solved by means of a finite-element (FE) technique. The tsunami is computed on two different grids formed by triangular elements, one covering the near-field around Stromboli and the other also including the island of Panarea. The simulations show that the main tsunamigenic potential of the slides is restricted to the first tens of seconds of their motion when they interact with the shallow-water coastal area, and that it diminishes drastically in deep water. The simulations explain how the tsunamis that are generated in the Sciara del Fuoco area, are able to attack the entire coastline of Stromboli with larger effects on the northern coast than on the southern. Strong refraction and bending of the tsunami fronts is due to the large near-shore bathymetric gradient, which is also responsible for the trapping of the waves and for the persistence of the oscillations. Further, the first tsunami produces large waves and runup heights comparable with the observations. The simulated second tsunami is only slightly smaller, though it was induced by a mass that is approximately one third of the first. The arrival of the first tsunami is negative, in accordance with most eyewitness reports. Conversely, the leading wave of the second tsunami is positive.

The landslides and tsunamis of the 30th of December 2002 in Stromboli analysed through numerical simulations

Three‐dimensional simulation of tsunami run‐up around conical island

. Solitary wave runup on a non-plane beach is studied analytically and numerically. For the theoretical approach, nonlinear shallow-water theory is applied to obtain the analytical solution for the simplified bottom geometry, such as an inclined channel whose cross-slope shape is parabolic. It generalizes Carrier-Greenspan approach for long wave runup on the inclined plane beach that is currently used now. For the numerical study, the Reynolds Averaged Navier-Stokes (RANS) system is applied to study soliton runup on an inclined beach and the detailed characteristics of the wave processes (water displacement, velocity field, turbulent kinetic energy, energy dissipation) are analyzed. In this study, it is theoretically and numerically proved that the existence of a parabolic cross-slope channel on the plane beach causes runup intensification, which is often observed in post-tsunami field surveys.

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Two- and three-dimensional computation of solitary wave runup on non-plane beach

This paper and its companion paper describe the development of a finite-element model based on modified Boussinesq equations and the applications of the model to harbor resonance problems. This first of the two papers reports the model development and validations. A Galerkin finite-element method with linear elements is employed in the model. Auxiliary variables are introduced to remove the spatial third derivative terms in the governing equations, and an implicit predictor-corrector iterative scheme is used in the time integration. The treatments of various boundary conditions, including the perfect reflecting boundary with an irregular geometry, the absorbing (sponge layer) boundary, and the incident wave boundary, are described. Numerical results are obtained for several examples, and their accuracy is checked by comparing numerical results with either experimental data or analytical solutions.

Finite-Element Model for Modified Boussinesq Equations. I: Model Development

The finite-element model for modified Boussinesq equations developed in Part I of this paper is used to study harbor resonance problems. The internal wavemaker technique is implemented on an unstructured finite-element mesh, and a sponge layer is used as an open boundary condition. Both linear and nonlinear harbor oscillations in rectangular and circular basins are examined, and experimental measurements and analytical solutions are used to validate the model. Finally, the wavelet transform is employed to analyze transient features of the nonlinear resonance problem.

Seung-Buhm Woo

Papers

Three‐dimensional simulation of tsunami run‐up around conical island

Two- and three-dimensional computation of solitary wave runup on non-plane beach

Finite-Element Model for Modified Boussinesq Equations. I: Model Development

Finite-Element Model for Modified Boussinesq Equations. II: Applications to Nonlinear Harbor Oscillations

A synchronously coupled tide–wave–surge model of the Yellow Sea