Numerical Simulation of Heat Loads in a Cryogenic H2/O2 Rocket Combustion Chamber

A Finite Element solver for a coupled simulation of fluid and structure in an axisymmetric domain is presented. The method employs an explicit solution of the flow and structure variables. The computational domain of the fluid is discretised by unstructured triangles and rectangles while the sturcture domain is discretised by unstructured triangles only. For the purpose of code validation the solution of in total three test cases are shown. One test case deals with the structure only while the other two simulate heat transfer problems with a defined temperature distribution along a boundary wall and coupled conditions. Finally the code is used to simulate the heat load in a cryogenic H₂/O₂ rocket combustion chamber.     

Numerical solution of a free boundary problem by continuation

An algorithm is described for the numerical solution of a free boundary problem using continuation. The problem considered is a quasi-steady problem arising in electrochemical machining. The domain of the problem is mapped onto a square and the governing nonlinear equations are discretised using finite differences. A Newton-like iteration is employed for the solution of the nonlinear algebraic system and global convergence is achieved by means of continuation. Numerical results are included.     

Nonlinear dynamics of endothelial cells

We propose a mathematical model that governs endothelial cell pattern formation on a biogel surface. The model accounts for diffusion and chemotactic motion of the cells, diffusion of the growth factor and effective biochemical reactions. The model admits a basic steady state that corresponds to a spatially uniform distribution of both the cells and the growth factor. We perform a weakly nonlinear stability analysis of the basic state in order to determine whether spatially nonuniform steady patterns can appear in the system when the basic state becomes unstable. The main results can be summarized as follows. No steady patterns can bifurcate from the basic state if the rate of decay of the growth factor is small. Increasing the rate of decay of the growth factor allows one to observe steady patterns, provided that diffusion of the growth factor is sufficiently slow. Specifically, the work focuses on the occurrence of hexagons and stripes. Most often hexagons are observed. In order for stripes to occur, the chemotactic sensitivity of the endothelial cells and/or their biochemical activity have to be reduced.     

Asymptotic Profile of Species Migrating on a Growing Habitat

This paper deals with a diffusive logistic equation on one dimensional isotropically growing domain. The model equation on growing domains is first presented, and the comparison principle is then proved. The asymptotic behavior of temporal solutions to the reaction-diffusion problem is given by constructing upper and lower solutions. Our result shows that when the domain grows slowly, the species successfully spreads to the whole habitat and stabilizes at a positive steady state, while it dies out in the long run if the domain grows fast. Numerical simulations are also presented to illustrate the analytical result.     

A DLM/FD/IB Method for Simulating Cell/Cell and Cell/Particle Interaction in Microchannels

A spring model is used to simulate the skeleton structure of the red blood cell （RBC） membrane and to study the red blood cell （RBC） rheology in Poiseuille flow with an immersed boundary method. The lateral migration properties of many cells in Poiseuille flow have been investigated. The authors also combine the above methodology with a distributed Lagrange multiplier/fictitious domain method to simulate the interaction of cells and neutrally buoyant particles in a microchannel for studying the margination of particles.     

The effects of coupling on pattern formation in a simple autocatalytic system

The spatiotetnporal structures that can arise in two identicalcells, each governed by cubic autocatalator kinetics and coupledvia the diffusive interchange of a reactant, are discussed.The coupling gives rise to five spatially uniform steady states,one of which exists in the uncoupled system. By studying thelinearized equations, it is found that three of these steadystates, including that of the uncoupled system, may give riseto the possibility of bifurcations to spatially nonuniform steadystates. In the case of the steady state corresponding to thatof the uncoupled system, it is seen that the coupling leadsto bifurcations not present in the uncoupled system which giverise to locally stable nonuniform steady states. A weakly nonlinearanalysis is developed for both small and large coupling strengtha, and for parameter values in a neighbourhood of the bifurcationpoints on the new steady states. This clarifies the nature ofthe nonuniform solutions close to bifurcation, which are thenfollowed numerically using a path-following technique. The couplingis found to produce extra nonuniform steady solutions whichare stable close to their bifurcation points.     

Lower Bounds and Nonuniform Time Discretization for Approximation of Stochastic Heat Equations

We study algorithms for approximation of the mild solution of stochastic heat equations on the spatial domain ]0, 1[^d. The error of an algorithm is defined in L₂-sense. We derive lower bounds for the error of every algorithm that uses a total of N evaluations of one-dimensional components of the driving Wiener process W. For equations with additive noise we derive matching upper bounds and we construct asymptotically optimal algorithms. The error bounds depend on N and d, and on the decay of eigenvalues of the covariance of W in the case of nuclear noise. In the latter case the use of nonuniform time discretizations is crucial.     

Simulation of short crack propagation using a hybrid boundary element technique

Service life of cyclically loaded components is often determined by the propagation of short fatigue cracks, which is highly influenced by microstructural features such as grain boundaries. A two-dimensional model to simulate the growth of such stage I-cracks is presented. The crack is discretised by dislocation discontinuity boundary elements and the direct boundary element method is used to mesh the grain boundaries. A superposition procedure couples these different boundary element methods to employ them in one model. Varying elastic properties of the grains are considered and their influence on short crack propagation is studied. A change in crack tip slide displacement determining short crack propagation is observed. (© 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A Porous Media Model to Describe the Behaviour of Brain Tissue

The human brain is a very sensitive organ. Even small changes in the cranium cavity can cause life–threatening effects. In case of medical intervention, biomechanics can assist the therapy decisions by simulating the physical behaviour of brain tissue, e.g., the coupled interaction of the fluid motion and the deformation of the brain tissue. In the context of the Theory of Porous Media (TPM), a convenient model of the brain is introduced, which is able to simulate essential mechanical effects in the porous structure of the brain material. The fluid–saturated brain can be treated as an immiscible binary mixture of constituents. In this macroscopic biphasic model, the mixture consists of a solid phase (brain tissue) and a fluid phase (interstitial fluid or blood plasma). Both constituents are assumed to be materially incompressible. The resulting set of coupled partial differential equations is then spatially discretised using mixed finite elements with a backward Euler time integration. Numerical examples are presented illustrating the fundamental effects on the brain tissue under heart–rate dependent pulsative pressure variations. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Adaptive space-time boundary element methods for the wave equation

For the solution of the wave equation a space-time energetic boundary integral formulation is used. The resulting single layer boundary integral equation is discretised by a conforming ansatz space on the lateral boundary. To derive an adaptive scheme an a posteriori error estimator based on the representation formula is used. Finally, numerical examples for a one-dimensional spatial domain are presented. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Numerical solution of thermal convection problems using the multidomain boundary element method

The multidomain dual reciprocity method (MD‐DRM) has been effectively applied to the solution of two‐dimensional thermal convection problems where the momentum and energy equations govern the motion of a viscous fluid. In the proposed boundary integral method the domain integrals are transformed into equivalent boundary integrals by the dual reciprocity approach applied in a subdomain basis. On each subregion or domain element the integral representation formulas for the velocity and temperature are applied and discretised using linear continuous boundary elements, and the equations from adjacent subregions are matched by additional continuity conditions. Some examples showing the accuracy, the efficiency and flexibility of the proposed method are presented. © 2002 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 18: 469–489, 2002; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/num.10016     

A comparison of solvers for linear complementarity problems arising from large-scale masonry structures

We compare the numerical performance of several methods for solving the discrete contact problem arising from the finite element discretisation of elastic systems with numerous contact points. The problem is formulated as a variational inequality and discretised using piecewise quadratic finite elements on a triangulation of the domain. At the discrete level, the variational inequality is reformulated as a classical linear complementarity system. We compare several state-of-art algorithms that have been advocated for such problems. Computational tests illustrate the use of these methods for a large collection of elastic bodies, such as a simplified bidimensional wall made of bricks or stone blocks, deformed under volume and surface forces. This work was supported by the Engineering and Physical Science Research Council of Great Britain under grant GR/S35101, and the first author was supported by a fellowship from the Royal Society of Edinburgh.     

Modeling of contaminant migration in groundwater: A continuum mechanical approach using in the theory of porous media

Over the past few decades, soil and subsequently groundwater contamination have raised growing concern as factors threatening the life of humans and other organisms as well as agricultural production sustainability. This study focuses on continuum mechanical multi-phase, multi-component formulation within the theory of porous media (TPM) to simulate the convective as well as diffusive driven contaminant tranport in groundwater including conversion processes. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Model of chemotaxis with threshold density and singular diffusion

A quasilinear singular parabolic system corresponding to recent models of chemotaxis in which (1) there is an impassable threshold for the density of cells and (2) the diffusion of cells becomes singular (fast or superdiffusion) when the density approaches the threshold. It is proved that for some range of parameters describing the relation between the diffusive and the chemotactic component of the cell flux there are global-in-time classical solutions which in some cases are separated from the threshold uniformly in time. Global-in-time weak solutions in the case of fast diffusion and the set of stationary states are studied as well. The applications of the general results to particular models are shown.     

Dynamic Wave Propagation in Porous Media Semi-Infinite Domains

The problem of dynamic wave propagation in semi-infinite domains is of great importance, especially, in subjects of applied mechanics and geomechanics, such as the issues of earthquake wave propagation in an infinite half-space and soil-structure interaction under seismic loading. In such problems, the elastic waves are supposed to propagate to infinity, which requires a special treatment of the boundaries in initial boundary-value problems (IBVP). Saturated porous materials, e. g. soil, basically represent volumetrically coupled solid-fluid aggregates. Based on the continuum-mechanical principles and the established macroscopic Theory of Porous Media (TPM) [1, 2], the governing balance equations yield a coupled system of partial differential equations (PDE). Restricting the discussion to the isothermal and geometrically linear case, this system comprises the solid and fluid momentum balances and the overall volume balance, and can be conveniently treated numerically following an implicit monolithic approach [3]. Therefore, the equations are firstly discretised in space using the mixed Finite Element Method (FEM) together with quasi-static Infinite Elements (IE) at the boundaries that represent the extension of the domain to infinity [4], and secondly in time using an appropriate implicit time-integration scheme. Additionally, a stable implementation of the Viscous Damping Boundary (VDB) method [5] for the simulation of transient waves at infinity is presented, which implicitly treats the damping boundary terms in a weakly imposed sense. The proposed algorithm is implemented into the FE tool PANDAS and tested on a two-dimensional IBVP. (© 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

具有非局部时滞的扩散Nicholson苍蝇方程的波前解

研究了具有非局部时滞的扩散Nicholson苍蝇方程,其中时滞由一个定义在所有过去时间和整个一维空间区域上的积分卷积表示.当时滞核是强生成核时,根据线性链式技巧和几何奇异扰动理论,获得了小时滞时波前解的存在性.     

Optimal Control of Multibody Dynamics with Contact

This work discusses two different structure preserving integrators in the framework of optimal control simulations with contact. The first one is a variational integrator, based on the constrained version of the Lagrange-D'Alembert. The resulting scheme preserves the symplecticity and the momentum maps of the simulated multibody dynamics. The second integrator is an energy momentum scheme and it is based on the augmented Hamiltonian equations, which are discretised using the discrete derivative in [2]. Both integrators are applied to simulate the optimal control of compass gait, for which the contact between the foot and the ground is modelled as perfectly plastic contact. The second example represents a monopedal jumper and it is used to examine the dynamical behaviour of the perfectly elastic and perfectly plastic contact formulation. The resulting differential algebraic equations (DAEs) are solved by the aforementioned symplectic momentum method. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A continuum model for free growth in living materials

We focus in this work on isotropically growing materials. An adaptive algorithm is used in order to maintain a stress-free state during growth if no external loads are applied, but keeping the volume growth defined by a former kinetic. The proposed model is based on a modified multiplicative split of the deformation gradient into a growth part and an elastic part. The growth part will be isotropic if the elastic deformations are favourable, otherwise the growth will find a more comfortable direction. Three-dimensional examples based on different kinetics are presented and discussed using the numerical model. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Phase field simulations on the interaction of ferroelectric domains with point defects

One of the suspected micro-mechanical mechanisms causing electric fatigue in ferroelectric materials is the hindering and blocking of domain wall movement. These blocking or pinning phenomena are thought to be due to point defects which interact with domain walls and applied external loads. A phase field model employing the spontaneous polarization as an order parameter is used to simulate the inhomogeneous material behavior. The coupled field equations are solved using the Finite Element Method. The influence of a stationary point defect on a domain wall is shown in a numerical simulation. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A Random Walk on Rectangles Algorithm

In this article, we introduce an algorithm that simulates efficiently the first exit time and position from a rectangle (or a parallelepiped) for a Brownian motion that starts at any point inside. This method provides an exact way to simulate the first exit time and position from any polygonal domain and then to solve some Dirichlet problems, whatever the dimension. This method can be used as a replacement or complement of the method of the random walk on spheres and can be easily adapted to deal with Neumann boundary conditions or Brownian motion with a constant drift. AMS 2000 Subject Classification 60C05, 65N