Numerical Mathematics and Advanced Applications ENUMATH 2015

We present the dual reciprocity boundary element method (DRBEM) solution of the system of equations which model magnetohydrodynamic (MHD) flow in a pipe with moving lid at low magnetic Reynolds number. The external magnetic field acts in the pipe-axis direction generating the electric potential. The solution is obtained in terms of stream function, vorticity and electric potential in the cross-section of the pipe, and the pipe axis velocity is also computed under a constant pressure gradient. It is found that fluid flow concentrates through the upper right corner forming boundary layers with the effect of moving lid and increased magnetic field intensity. Electric field behavior is changed accordingly with the insulated and conducting portions of the pipe walls. Fluid moves in the pipe-axis direction with an increasing rate of magnitude when Hartmann number increases. The boundary only nature of DRBEM provides the solution at a low computational expense.

A numerical investigation of unsteady, two-dimensional double diffusive convection flow through a lid-driven square enclosure is carried on. The left and bottom walls of the enclosure are either uniformly or non-uniformly heated and concentrated, while the right vertical wall is maintained at a constant cold temperature. The top wall is insulated and it moves to the right with a constant velocity. The numerical solution of the coupled nonlinear differential equations is based on the use of dual reciprocity boundary element method (DRBEM) in spatial discretization and an unconditionally stable backward implicit finite difference scheme for the time integration. Due to the coupling and the nonlinearity, an iterative process is employed between the equations. The boundary only nature of the DRBEM and the use of the fundamental solution of Laplace equation make the solution process computationally easier and less expensive compared to other domain discretization methods. The study focuses on the effects of uniform and non-uniform heating and concentration of the walls for various values of physical parameters on the double-diffusive convection in terms of streamlines, isotherms and isoconcentration lines.

We present an extension of the complete flux scheme for conservation laws containing a linear source. In our new scheme, we split off the linear part of the source and incorporate this term in the homogeneous flux, the remaining nonlinear part is included in the inhomogeneous flux. This approach gives rise to modified homogeneous and inhomogeneous fluxes, which reduce to the classical fluxes for vanishing linear source. On the other hand, if the linear source is large, the solution of the underlying boundary value problem is oscillatory, resulting in completely different numerical fluxes. We demonstrate the performance of the homogeneous flux approximation.

The today’s demands for simulation and optimization tools for water supply networks are permanently increasing. Practical computations of large water supply networks show that rather small time steps are needed to get sufficiently good approximation results – a typical disadvantage of low order methods. Having this application in mind we use higher order time discretizations to overcome this problem. Such discretizations can be achieved using so-called strong stability preserving Runge-Kutta methods which are especially designed for hyperbolic problems. We aim at approximating entropy solutions and are interested in weak solutions and variational formulations. Therefore our intention is to compare different space discretizations mostly based on variational formulations, and combine them with a second-order two-stage SDIRK method. In this paper, we will report on first numerical results considering scalar hyperbolic conservation laws.

We present a flux approximation scheme for the incompressible Navier-Stokes equations, that is based on a flux approximation scheme for the scalar advection-diffusion-reaction equation that we developed earlier. The flux is computed from local boundary value problems (BVPs) and is expressed as a sum of a homogeneous and an inhomogeneous part. The homogeneous part depends on the balance of the convective and viscous forces and the inhomogeneous part depends on source terms included in the local BVP.

Recently, we showed in (O. Kolb, SIAM J. Numer. Anal., 52 (2014), pp. 2335–2355) for which parameter range the compact third order WENO reconstruction procedure introduced in (D. Levy, G. Puppo, and G. Russo, SIAM J. Sci. Comput., 22 (2000), pp. 656–672) reaches the optimal order of accuracy (h3 in the smooth case and h2 near discontinuities). This is the case for the parameter choice ɛ = Khq in the weight design with q ≤ 3 and pq ≥ 2, where p ≥ 1 is the exponent used in the computation of the weights in the WENO scheme. While these theoretical results for the convergence rates of the WENO reconstruction procedure could also be validated in the numerical tests, the application within the semi-discrete central scheme of (A. Kurganov, and D. Levy, SIAM J. Sci. Comput., 22 (2000), pp. 1461–1488) together with a third order TVD-Runge-Kutta scheme for the time integration did not yield a third order accurate scheme in total for q > 2. The aim of this follow-up paper is to explain this observation with further analytical and numerical results.

The problem in dimensionless variables reduces to three systems of equations for the stream function and the electric potential in three regions of a strip (the rectangular domain bounded by the electrodes and two half-strips). The singular integral equations obtained from the integral representation of the solutions and the matching conditions are disctretized and a linear system of algebraic equations is obtained. The velocity, the electric field and the generator power are calculated.

We are interested in the numerical solution of linear hyperbolic problems using continuous finite elements of arbitrary order. It is well known that this kind of methods, once the weak formulation has been written, leads to a system of ordinary differential equations in $$\mathbb{R}^{N}$$ , where N is the number of degrees of freedom. The solution of the resulting ODE system involves the inversion of a sparse mass matrix that is not block diagonal. Here we show how to avoid this step, and what are the consequences of the choice of the finite element space. Numerical examples show the correctness of our approach.

The purpose of the paper is to analyse the effect of hp mesh adaptation when discretized versions of finite element mixed formulations are applied to elliptic problems with singular solutions. Two stable configurations of approximation spaces, based on affine triangular and quadrilateral meshes, are considered for primal and dual (flux) variables. When computing sufficiently smooth solutions using regular meshes, the first configuration gives optimal convergence rates of identical approximation orders for both variables, as well as for the divergence of the flux. For the second configuration, higher convergence rates are obtained for the primal variable. Furthermore, after static condensation is applied, the condensed systems to be solved have the same dimension in both configuration cases, which is proportional to their border flux dimensions. A test problem with a steep interior layer is simulated, and the results demonstrate exponential rates of convergence. Comparison of the results obtained with H1-conforming formulation are also presented.

This work is devoted to the Directional Do-Nothing (DDN) condition as an outflow boundary condition for the incompressible Navier-Stokes equation. In contrast to the Classical Do-Nothing (CDN) condition, we have stability, existence of weak solutions and, in the case of small data, also uniqueness. We derive an a priori error estimate for this outflow condition for finite element discretizations with inf-sup stable pairs. Stabilization terms account for dominant convection and the divergence free constraint. Numerical examples demonstrate the stability of the method.

We consider Galerkin approximation in space of linear parabolic initial-boundary value problems where the elliptic operator is symmetric and thus induces an energy norm. For two related variational settings, we show that the quasi-optimality constant equals the stability constant of the L2-projection with respect to that energy norm.

Projection based variational multiscale (VMS) methods are a very successful technique in the numerical simulation of high Reynolds number flow problems using coarse discretizations. However, their implementation into an existing (legacy) codes can be very challenging in practice. We propose a second order variant of projection-based VMS method for non-isothermal flow problems. The method adds stabilization as a decoupled post-processing step for both velocity and temperature, and thus can be efficiently and easily used with existing codes. In this work, we propose the algorithm and give numerical results for convergence rates tests and coarse mesh simulation of Marsigli flow.

This paper gives numerical experiments for the Finite Element Heterogeneous Multiscale Method applied to problems in linear elasticity, which has been analyzed in Abdulle (Math Models Methods Appl Sci 16:615–635, 2006). The main results for the FE-HMM a priori errors are stated and their sharpness are verified though numerical experiments.

We introduce an abstract concept for decomposing spaces with respect to a substructuring of a bounded domain. In this setting we define weakly conforming finite element approximations of quadratic minimization problems. Within a saddle point approach the reduction to symmetric positive Schur complement systems on the skeleton is analyzed. Applications include weakly conforming variants of least squares and minimal residuals.

In this work we present an iterative coupling scheme for the quasi-static Biot system of poroelasticity. For the discretization of the subproblems describing mechanical deformation and single-phase flow space-time finite element methods based on a discontinuous Galerkin approximation of the time variable are used. The spatial approximation of the flow problem is done by mixed finite element methods. The stability of the approach is illustrated by numerical experiments. The presented variational space-time framework is of higher order accuracy such that problems with high fluctuations become feasible. Moreover, it offers promising potential for the simulation of the fully dynamic Biot–Allard system coupling an elastic wave equation for solid’s deformation with single-phase flow for fluid infiltration.

This paper is concerned with the numerical solution of dynamic elasticity by the discontinuous Galerkin (dG) method. We consider the linear and nonlinear St. Venant-Kirchhoff model. The dynamic elasticity problem is split into two systems of first order in time. They are discretized by the discontinuous Galerkin method in space and backward difference formula in time. The developed method is tested by numerical experiments. Then the method is combined with the space-time dG method for the solution of compressible flow in a time dependent domain and used for the numerical simulation of fluid-structure interaction.

We propose a modified local discontinuous Galerkin (LDG) method for second–order elliptic problems that does not require extrinsic penalization to ensure stability. Stability is instead achieved by showing a discrete Poincaré–Friedrichs inequality for the discrete gradient that employs a lifting of the jumps with one polynomial degree higher than the scalar approximation space. Our analysis covers rather general simplicial meshes with the possibility of hanging nodes.

We develop an adaptive method of time layers with a linearly implicit Rosenbrock method as time integrator and symmetric interior penalty Galerkin method for space discretization for the advective Allen-Cahn equation with a non-divergence-free velocity field. Numerical simulations for advection-dominated problems demonstrate the accuracy and efficiency of the adaptive algorithm for resolving the sharp layers occurring in interface problems with small surface tension.

We present the numerical solution of a hydrodynamics model of flocking using a suitable modified semi-implicit discontinuous Galerkin method. The investigated model describing the dynamics of flocks of birds or other individual entities forming herds or swarms was introduced by Fornasier et al. (Physica D 240(1):21–31, 2011). The main idea of this model comes from the well known Cucker-Smale model. The resulting equations consist of the Euler equations for compressible flow with an additional non-local non-linear source term. The model is discretized by the semi-implicit discontinuous Galerkin method for the compressible Euler equations of Feistauer and Kučera (J Comput Phys 224(1):208–221, 2007). We show that with a suitable treatment of the source term we can use this method for models like the model of flocking and find a numerical solution very efficiently.

In this work, we introduce discontinuous Galerkin and enriched Galerkin formulations for the spatial discretization of phase-field fracture propagation. The nonlinear coupled system is formulated in terms of the Euler-Lagrange equations, which are subject to a crack irreversibility condition. The resulting variational inequality is solved in a quasi-monolithic way in which the irreversibility condition is incorporated with the help of an augmented Lagrangian technique. The relaxed nonlinear system is treated with Newton’s method. Numerical results complete the present study.

We present a numerical method for the solution of integro-differential equations describing motion of an incompressible viscoelastic fluid with memory. In particular, the system of equations consists of the momentum conservation equation with the Cauchy stress tensor divided in a viscous and an elastic parts which depend non-linearly on the symmetric part of velocity gradient and non-linearly on the past values of the Finger strain tensor, respectively. The momentum conservation equation is completed with system of equations that describes relation between the velocity gradient and the Finger strain tensor. The method is based on a discontinuous Galerkin method in the spatial variables and the BDF methods in the time variables.

In this paper we investigate the stability of the space-time discontinuous Galerkin method (STDGM) for the solution of nonstationary, linear convection-diffusion-reaction problem in time-dependent domains formulated with the aid of the arbitrary Lagrangian-Eulerian (ALE) method. At first we define the continuous problem and reformulate it using the ALE method, which replaces the classical partial time derivative with the so called ALE-derivative and an additional convective term. In the second part of the paper we discretize our problem using the space-time discontinuous Galerkin method. The space discretization uses piecewise polynomial approximations of degree p ≥ 1, in time we use only piecewise linear discretization. Finally in the third part of the paper we present our results concerning the unconditional stability of the method.

We apply continuous and discontinuous Galerkin time discretization together with standard finite element method for space discretization to the heat equation. For the numerical solution arising from these discretizations we present a guaranteed and fully computable a posteriori error upper bound. Moreover, we present local asymptotic efficiency estimate of this bound.

The reduced reliability of next generation exascale systems means that the resiliency properties of a numerical algorithm will become an important factor in both the choice of algorithm, and in its analysis. The multigrid algorithm is the workhorse for the distributed solution of linear systems but little is known about its resiliency properties and convergence behavior in a fault-prone environment. In the current work, we propose a probabilistic model for the effect of faults involving random diagonal matrices. We summarize results of the theoretical analysis of the model for the rate of convergence of fault-prone multigrid methods which show that the standard multigrid method will not be resilient. Finally, we present a modification of the standard multigrid algorithm that will be resilient.

A variant of a nonlinear FETI-DP domain decomposition method is considered. It is combined with a parallel algebraic multigrid method (BoomerAMG) in a way which completely removes sparse direct solvers from the algorithm. Scalability to 524,288 MPI ranks is shown for linear elasticity and nonlinear hyperelasticity using more than half of the JUQUEEN supercomputer (JSC, Jülich; TOP500 rank: 11th).

We present a parallel and efficient multilevel solution method for the nonlinear systems arising from the discretization of Navier–Stokes (N-S) equations with finite differences. In particular we study the incompressible, unsteady N-S equations with periodic boundary condition in time. A sequential time integration limits the parallelism of the solver to the spatial variables and can therefore be an obstacle to parallel scalability. Time periodicity allows for a space-time discretization, which adds time as an additional direction for parallelism and thus can improve parallel scalability. To achieve fast convergence, we used a space-time multigrid algorithm with a SCGS smoothing procedure (symmetrical coupled Gauss–Seidel, a.k.a. box smoothing). This technique, proposed by Vanka (J Comput Phys 65:138–156, 1986), for the steady viscous incompressible Navier–Stokes equations is extended to the unsteady case and its properties are studied using local Fourier analysis. We used numerical experiments to analyze the scalability and the convergence of the solver, focusing on the case of a pulsatile flow.

Conservation laws of advection-diffusion-reaction (ADR) type are ubiquitous in continuum physics. In this paper we outline discretization of these problems and iterative schemes for the resulting linear system. For discretization we use the finite volume method in combination with the complete flux scheme. The numerical flux is the superposition of a homogeneous flux, corresponding to the advection-diffusion operator, and the inhomogeneous flux, taking into account the effect of the source term (ten Thije Boonkkamp and Anthonissen, J Sci Comput 46(1):47–70, 2011). For a three-dimensional conservation law this results in a 27-point coupling for the unknown as well as the source term. Direct solution of the sparse linear systems that arise in 3D ADR problems is not feasible due to fill-in. Iterative solution of such linear systems remains to be the only efficient alternative which requires less memory and shorter time to solution compared to direct solvers. Iterative solvers require a preconditioner to reduce the number of iterations. We present results using several different preconditioning techniques and study their effectiveness.

Evolutionary processes arise in many areas of applied mathematics, however since the solution at any time depends on the solution at previous time steps, these types of problems are inherently difficult to parallelize. In this paper, we make a simple proposal of a parallel approach for the solution of evolutionary problems with implicit time step schemes. We derive and demonstrate our approach for both the linear diffusion equation and the convection-diffusion equation. Using an all-at-once approach, we solve for all time steps simultaneously using a parallelizable over time preconditioner within a standard iterative method.

Discretization of (linearized) convection-diffusion-reaction problems yields a large and sparse non symmetric linear system of equations, 1 $$\displaystyle{ A\mathbf{x} = \mathbf{b}. }$$ In this work, we compare the computational behavior of the Induced Dimension Reduction method (IDR(s)) (Sonneveld and van Gijzen, SIAM J Sci Comput 31(2):1035–1062, 2008), with other short-recurrences Krylov methods, specifically the Bi-Conjugate Gradient Method (Bi-CG) (Fletcher, Conjugate gradient methods for indefinite systems. In: Proceedings of the Dundee conference on numerical analysis, pp 73–89, 1976), restarted Generalized Minimal Residual (GMRES(m)) (Saad and Schultz, SIAM J Sci Stat Comput 7:856–869, 1986), and Bi-Conjugate Gradient Stabilized method (Bi-CGSTAB) (van der Vorst, SIAM J Sci Stat Comput 13(2):631–644, 1992).

In the present study, we establish two new block variants of the Conjugate Orthogonal Conjugate Gradient (COCG) and the Conjugate A-Orthogonal Conjugate Residual (COCR) Krylov subspace methods for solving complex symmetric linear systems with multiple right hand sides. The proposed Block iterative solvers can fully exploit the complex symmetry property of coefficient matrix of the linear system. We report on extensive numerical experiments to show the favourable convergence properties of our newly developed Block algorithms for solving realistic electromagnetic simulations.

In this contribution we are concerned with efficient model reduction for multiscale problems arising in lithium-ion battery modeling with spatially resolved porous electrodes. We present new results on the application of the reduced basis method to the resulting instationary 3D battery model that involves strong non-linearities due to Buttler-Volmer kinetics. Empirical operator interpolation is used to efficiently deal with this issue. Furthermore, we present the localized reduced basis multiscale method for parabolic problems applied to a thermal model of batteries with resolved porous electrodes. Numerical experiments are given that demonstrate the reduction capabilities of the presented approaches for these real world applications.

In this paper we provide a general framework for model reduction methods applied to fluid flow in porous media. Using reduced basis and numerical homogenization techniques we show that the complexity of the numerical approximation of Stokes flow in heterogeneous media can be drastically reduced. The use of such a computational framework is illustrated at several model problems such as two and three scale porous media.

The Reduced Basis Method (RBM) (Rozza et al., Archiv Comput Methods Eng 15:229–275, 2008) generates low-order models for efficient evaluation of parametrized PDEs in many-query and real-time contexts. It can be seen as a parametric model reduction method (Benner et al., SIAM Rev 57(4):483–531, 2015), where greedy selection is combined with a projection space composed of solution snapshots. The approximation quality is certified by using rigorous error estimators. We apply the RBM to systems of Maxwell’s equations arising from electrical circuits. Using microstrip models, the input-output behaviour of the interconnect structures is approximated for a certain frequency range. Typically, an output is given by a linear functional, but in the case of impedance parameters (also called Z-parameters), the output is quadratic. An expanded formulation is used to rewrite the system to compliant form, i.e., a form, where the input and output are identical. This enables fast convergence in the approximation error and thus very low reduced model sizes. A numerical example from the microwave regime shows the advantage of this approach.

A (multi-)wavelet expansion is used to derive a rigorous bound for the (dual) norm Reduced Basis residual. We show theoretically and numerically that the error estimator is online efficient, reliable and rigorous. It allows to control the exact error (not only with respect to a “truth” discretization).

We developed a reduced order model (ROM) using the proper orthogonal decomposition (POD) to compute efficiently the labyrinth and spot like patterns of the FitzHugh-Nagumo (FNH) equation. The FHN equation is discretized in space by the discontinuous Galerkin (dG) method and in time by the backward Euler method. Applying POD-DEIM (discrete empirical interpolation method) to the full order model (FOM) for different values of the parameter in the bistable nonlinearity, we show that using few POD and DEIM modes, the patterns can be computed accurately. Due to the local nature of the dG discretization, the POD-DEIM requires less number of connected nodes than continuous finite element for the nonlinear terms, which leads to a significant reduction of the computational cost for dG POD-DEIM.

A method is proposed for constructing local parametrizations of orthogonal bases and of subspaces by computing trajectories in the Stiefel and the Grassmann manifold, respectively. The trajectories are obtained by exploiting sensitivity information on the singular value decomposition with respect to parametric changes and a Taylor-like local linearization suitably adapted to the underlying manifold structure. An important practical application of the proposed approach is parametric model reduction (pMOR). The connection with pMOR is discussed in detail and the results are illustrated by numerical experiment.

In this paper a reduced-order strategy is applied to solve a multiobjective optimal control problem governed by semilinear parabolic partial differential equations. These problems often arise in practical applications, where the quality of the system behaviour has to be measured by more than one criterium. The weighted sum method is exploited for defining scalar-valued nonlinear optimal control problems built by introducing additional optimization parameters. The optimal controls corresponding to specific choices of the optimization parameters are efficiently computed by the reduced-basis method. The accuracy is guaranteed by an a-posteriori error estimate.

In this work, a framework for coupling arbitrary Lagrangian-Eulerian fluid-structure interaction with phase-field fracture is suggested. The key idea is based on applying the weak form of phase-field fracture, including a crack irreversibility constraint, to the nonlinear coupled system of Navier-Stokes and elasticity. The resulting setting is formulated via variational-monolithic coupling and has four unknowns: velocities, displacements, pressure, and a phase-field variable. The inequality constraint is imposed through penalization using an augmented Lagrangian algorithm. The nonlinear problem is solved with Newton’s method. The framework is tested in terms of a numerical example in which computational stability is demonstrated by evaluating goal functionals on different spatial meshes.

In this paper we consider homogeneous Dirichlet problem for the Lamé system with singularity caused by the reentrant corner to the boundary of the two-dimensional domain. For this problem we define the solution as a R _ν -generalized one; we state its existence and uniqueness in the weighted set $$\mathring{\mathbf{W}}_{2,\nu }^{1}(\varOmega,\delta )$$ . On the basis of the R _ν -generalized solution we construct weighted finite element method. We prove that the approximate solution converges to the exact one with the rate O(h) in the norm of W_2, ν 1(Ω), and results of numerical experiments are presented.

In this paper, we show a local a priori error estimate for the Poisson equation in three space dimensions (3D), where the source term is a Dirac measure concentrated on a line. This type of problem can be found in many application areas. In medical engineering, e.g., blood flow in capillaries and tissue can be modeled by coupling Poiseuille’s and Darcy’s law using a line source term. Due to the singularity induced by the line source term, finite element solutions converge suboptimal in classical norms. However, quite often the error at the singularity is either dominated by model errors (e.g. in dimension reduced settings) or is not the quantity of interest (e.g. in optimal control problems). Therefore we are interested in local error estimates, i.e., we consider in space a L2-norm on a fixed subdomain excluding a neighborhood of the line, where the Dirac measure is concentrated. It is shown that linear finite elements converge optimal up to a log-factor in such a norm. The theoretical considerations are confirmed by some numerical tests.

We consider a multirate iterative scheme for the quasi-static Biot equations modelling the coupled flow and geomechanics in a porous medium. The iterative scheme is based on undrained splitting where the flow and mechanics equations are decoupled with the mechanics solve followed by the pressure solve. The multirate scheme proposed here uses different time steps for the two equations, that is, uses q flow steps for each coarse mechanics step and may be interpreted as using a regularization parameter for the mechanics equation. We prove the convergence of the scheme and the proof reveals the appropriate regularization parameter and also the effect of the number of flow steps within coarse mechanics step on the convergence rate.

This work deals with numerical simulation of incompressible flow over a profile vibrating with two degrees of freedom. The profile can oscillate around prescribed elastic axis and vibrate in vertical direction and its motion is induced by the flow. The finite volume method was chosen for the solution, namely the so called Modified Causon’s Scheme, which is derived from TVD form of the classical predictor-corrector MacCormack scheme and enhanced with the use of the Arbitrary Lagrangian-Eulerian method in order to simulate unsteady flows. Various initial settings are considered (different inlet velocities, initial deviation angles and shifts in vertical direction). Stiffness is modelled both as linear and non-linear. Obtained results are compared with NASTRAN analysis (Čečrdle and Maleček, Verification FEM model of an aircraft construction with two and three degrees of freedom. Technical report R-3418/02, Aeronautical Research and Test Establishment, Prague, Letňany, 2002. In Czech). The resulting critical velocities for unstable oscillations are in the same interval for all simulated cases.

The two-dimensional, laminar, unsteady natural convection flow of a micropolar nanofluid (Al₂O₃-water) in a square enclosure under the influence of a magnetic field, is solved numerically using the Chebyshev spectral collocation method (CSCM). The nanofluid is considered as Newtonian and incompressible, and the nanoparticles and water are assumed to be in thermal equilibrium. The governing equations in nondimensional form are given in terms of stream function, vorticity, micrototaion and temperature. The coupled and nonlinear equations are solved iteratively in the time direction, and an implicit backward difference scheme is employed for the time integration. The boundary conditions of vorticity are computed within this iterative process using a CSCM discretization of the stream function equation. The main advantages of CSCM, such as the high accuracy and the ease of implementation, are made used of to obtain solutions for very high values of Ra and Ha, up to 107 and 1000, respectively.

In this study, possibility of reducing drag in turbulent pipe flow via phase randomization is investigated. Phase randomization is a passive drag reduction mechanism, the main idea behind which is, reduction in drag can be obtained via distrupting the wave-like structures present in the flow. To facilitate the investigation flow in a circular cylindrical pipe is simulated numerically. DNS (direct numerical simulation) approach is used with a solenoidal spectral formulation, hence the continuity equation is automatically satisfied (Tugluk and Tarman, Acta Mech 223(5):921–935, 2012). Simulations are performed for flow driven by a constant mass flux, at a bulk Reynolds number (Re) of 4900. Legendre polynomials are used in constructing the solenoidal basis functions employed in the numerical method.

In this study we deal with a problem of particulate matter dispersion modelling in a presence of a vegetation. We present a method to evaluate the efficiency of the barrier and to optimize its parameters. We use a CFD solver based on the RANS equations to model the air flow in a simplified 2D domain containing a vegetation block adjacent to a road, which serves as a source of the pollutant. Modelled physics captures the processes of a gravitational settling of the particles, dry deposition of the particles on the vegetation, turbulence generation by the road traffic and effect of the vegetation on the air flow. To optimize the effectivity of the barrier we employ a gradient based optimization process. The results show that the optimized variant relies mainly on the effect of increased turbulent diffusion by a sparse vegetation and less on the dry deposition of the pollutant on the vegetation.

The aim of this contribution is to present a new Newton-type solver for yield stress fluids, for instance for viscoplastic Bingham fluids. In contrast to standard globally defined (‘outer’) damping strategies, we apply weighting strategies for the different parts inside of the resulting Jacobian matrices (after discretizing with FEM), taking into account the special properties of the partial operators which arise due to the differentiation of the corresponding nonlinear viscosity function. Moreover, we shortly discuss the corresponding extension to fluids with a pressure-dependent yield stress which are quite common for modelling granular material. From a numerical point of view, the presented method can be seen as a generalized Newton approach for non-smooth problems.

This paper is interested in the numerical simulation of steady flows of laminar incompressible viscous and viscoelastic fluids through the channel with T-junction. The flow is described by the system of generalized incompressible Navier-Stokes equations. For the different choice of fluids model the different model of the stress tensor is used, Newtonian and Oldroyd-B models. Numerical tests are performed on three dimensional geometry, a branched channel with one entrance and two outlet parts. Numerical solution of the described models is based on cell-centered finite volume method using explicit Runge-Kutta time integration.

Fractional flow reserve (FFR) is the golden standard for making decision on surgical treatment of coronary vessels with multiple stenosis. Clinical measurements of FFR require expensive invasive procedure with endovascular ultrasound probe. In this work a method of FFR simulation is considered. It is based on modelling 1D haemodynamics in patient-specific coronary vessels network reconstructed from CT scans. In contrast to our previous studies we used explicit minimum oscillating 2nd order characteristic method for internal nodes and 2nd order approximation of compatibility conditions for discretization of boundary conditions in junctions. The model is applied for simulating the change of FFR due to variability of the vessels elasticity and autoregulation response rate.

Monocytes play a significant role in the atherosclerosis development. During the inflammation process, monocytes that circulate in the blood stream are activated. Upon activation, they adhere to the endothelium and extravasate through the latter to migrate into the intima. In this work we are concerned with the transmigration stage. Micropipette aspiration experiments show that monocytes behave as polymeric drops during suction. In our study, the constitutive equations for Oldroyd-B fluids are used to capture the viscoelastic behavior of monocytes. We first establish and analyze a simplified mathematical model describing the coupled deformation-flow of an individual monocyte in a microchannel. Then we describe the numerical implementation of the mathematical model using the level set method and show the numerical results. Further extensions of this model are also discussed.

Parallel overlapping Schwarz preconditioners are considered and applied to the structural block in monolithic fluid-structure interaction (FSI). The two-level overlapping Schwarz method uses a coarse level based on energy minimizing functions. Linear elastic as well as nonlinear, anisotropic hyperelastic structural models are considered in an FSI problem of a pressure wave in a tube. Using our recent parallel implementation of a two-level overlapping Schwarz preconditioner based on the Trilinos library, the total computation time of our FSI benchmark problem was reduced by more than a factor of two compared to the algebraic one-level overlapping Schwarz method used previously. Finally, also strong scalability for our FSI problem is shown for up to 512 processor cores.

We present a mathematical and numerical model for non-isothermal, compressible flow of a mixture of two ideal gases subject to gravity. This flow is described by the balance equations for mass, momentum and energy that are solved numerically by the scheme based on the method of lines. The spatial discretization is carried out by means of the finite volume method, where the staggered arrangement of variables is employed. The time integration is realized by the Runge-Kutta-Merson method. The article also contains test results obtained by the presented numerical scheme.

This paper presents results of computational studies of the evolution law for the constrained mean curvature flow. The considered motion law originates in the theory of phase transitions in crystalline materials. It describes the evolution of closed embedded curves with constant enclosed area. In the paper, the motion law is treated by the parametric method, which leads into the system of degenerate parabolic equations for the parametric description of the curve. This system is numerically solved by means of the flowing finite volume method enhanced by tangential redistribution. Qualitative and quantitative results of computational studies are presented.

The goal of this work is the analysis of a time-harmonic eddy current model with prescribed current intensities imposed on the boundary of the conducting domain. We will study a $$\boldsymbol{T},\phi -\phi$$ formulation, which combines a current vector potential $$\boldsymbol{T}$$ with a scalar potential ϕ. A significant advantage of this method is that the expensive vector unknown $$\boldsymbol{T}$$ has to be computed only in conductors. Moreover, the proposed numerical method avoids the building of cutting surfaces what is very convenient in the case of complex geometries.

We consider the time-dependent magnetic induction model as a step towards the resistive magnetohydrodynamics (MHD) model in incompressible media. Conforming nodal-based finite element (FE) approximations of the induction model with Taylor-Hood type FE as well as equal-order FE for the magnetic field and the magnetic pseudo-pressure are investigated. We consider a stabilized nodal-based FEM for the numerical solution. Error estimates are given for the semidiscrete model in space. Finally, we present results for the magnetic flux expulsion problem.

A finite element (FE) model, that explicitly discretizes a single 3D spherulite is proposed. A spherulite is a two-phase microstructure consisting of amorphous and crystalline regions. Crystalline regions, that grow from a central nucleus in the form of lamellae, have particular lattice orientations. In the FE analyses, 8-chain and crystal viscoplasticity constitutive models are employed. Stress-strain distributions and slip system activities in the spherulite microstructure are studied and found to be in good agreement with the literature. Influences of the crystallinity ratio on the yield stress and the initial Young’s modulus are also investigated.

This paper deals with flow induced vibrations of an elastic body. A simplified model of the human vocal fold is mathematically described. In order to consider the time dependent domain the arbitrary Lagrangian-Eulerian method is used. The viscous incompressible fluid flow and linear elasticity models are considered. The developed numerical schemes for the fluid flow and the elastic body are implemented by the in-house developed solver based on the finite element method. Preliminary numerical results testing the convergence of solver are presented.

We present a comparative study of integral operators used in nonlocal problems. The size of nonlocality is determined by the parameter δ. The authors recently discovered a way to incorporate local boundary conditions into nonlocal problems. We construct two nonlocal operators which satisfy local homogeneous Neumann boundary conditions. We compare the bulk and boundary behaviors of these two to the operator that enforces nonlocal boundary conditions. We construct approximations to each operator using perturbation expansions in the form of Taylor polynomials by consistently keeping the size of expansion neighborhood equal to δ. In the bulk, we show that one of these two operators exhibits similar behavior with the operator that enforces nonlocal boundary conditions.

European basket options are priced by solving the multi-dimensional Black–Scholes–Merton equation. Standard numerical methods to solve these problems often suffer from the “curse of dimensionality”. We tackle this by using a dimension reduction technique based on a principal component analysis with an asymptotic expansion. Adaptive finite differences are used for the spatial discretization. In time we employ a discontinuous Galerkin scheme. The efficiency of our proposed method to solve a five-dimensional problem is demonstrated through numerical experiments and compared with a Monte-Carlo method.

We consider second order hyperbolic equation with nonlocal integral boundary conditions. We study the spectrum of the weighted difference operator for the formulated problem. Using the characteristic function we investigate the spectrum of the transition matrix of the three-layered finite difference scheme and obtain spectral stability conditions subject to boundary parameters γ₀, γ₁ and piecewise constant weight functions.

In this paper, a Riemannian BFGS method is defined for minimizing a smooth function on a Riemannian manifold endowed with a retraction and a vector transport. The method is based on a Riemannian generalization of a cautious update and a weak line search condition. It is shown that, the Riemannian BFGS method converges (i) globally to a stationary point without assuming that the objective function is convex and (ii) superlinearly to a nondegenerate minimizer. The weak line search condition removes completely the need to consider the differentiated retraction. The joint diagonalization problem is used to demonstrate the performance of the algorithm with various parameters, line search conditions, and pairs of retraction and vector transport.

Convection is an important transport mechanism in physics. Especially, in fluid dynamics at high Reynolds numbers this term dominates. Modern mimetic discretization methods consider physical variables as differential k-forms and their discrete analogues as k-cochains. Convection, in this parlance, is represented by the Lie derivative, $$\mathcal{L}_{X}$$ . In this paper we design reduction operators, $$\mathcal{R}$$ from differential forms to cochains and define a discrete Lie derivative, L _X which acts on cochains such that the commutation relation $$\mathcal{R}\mathcal{L}_{X} = \mathsf{L}_{X}\mathcal{R}$$ holds.