• Abstract

A parametric reduced order model based on proper orthogonal decomposition with Galerkin projection has been developed and applied for the modeling of heat transport in T-junction pipes which are widely found in nuclear power reactor cooling systems. Thermal mixing of different temperature coolants in T-junction pipes leads to temperature fluctuations and this could potentially cause thermal fatigue in the pipe walls. The novelty of this paper is the development of a parametric ROM considering the three dimensional, incompressible, unsteady Navier-Stokes equations coupled with the heat transport equation in a finite volume regime. Two different parametric cases are presented in this paper: parametrization of the inlet temperatures and parametrization of the kinematic viscosity. Different training spaces are considered and the results are compared against the full order model. The first test case results to a computational speed-up factor of 374 while the second test case to one of 211.

• Abstract

An efficient algorithm is proposed for Bayesian model calibration, which is commonly used to estimate the model parameters of non-linear, computationally expensive models using measurement data. The approach is based on Bayesian statistics: using a prior distribution and a likelihood, the posterior distribution is obtained through application of Bayes' law. Our novel algorithm to accurately determine this posterior requires significantly fewer discrete model evaluations than traditional Monte Carlo methods. The key idea is to replace the expensive model by an interpolating surrogate model and to construct the interpolating nodal set maximizing the accuracy of the posterior. To determine such a nodal set an extension to weighted Leja nodes is introduced, based on a new weighting function. We prove that the convergence of the posterior has the same rate as the convergence of the model. If the convergence of the posterior is measured in the Kullback–Leibler divergence, the rate doubles. The algorithm and its theoretical properties are verified in three different test cases: analytical cases that confirm the correctness of the theoretical findings, Burgers' equation to show its applicability in implicit problems, and finally the calibration of the closure parameters of a turbulence model to show the effectiveness for computationally expensive problems.

• Abstract

We examine the meandering instability for prismatic lattices formed from triangles, squares and hexagons using a nearest neighbor kinetic Monte Carlo model. In the first two cases, which are Bravais lattices, we find that facets with the orientation favored in the equilibrium shape of isolated islands are most prone to this instability, while the analogous facet for the hexagonal lattice is the least unstable. We argue that this is due to a significant difference in the reconstructed/equilibrium versus the non-reconstructed edge energy for non-Bravais crystals. Surface/edge energy is typically modeled as a single-valued function of orientation. We put forward a simple geometric argument that suggests this picture is inadequate for crystals with a non-Bravais lattice structure. In the case of a hexagonally structured lattice, our arguments indicate that the non-reconstructed edge energy can be viewed as both discontinuous and multi-valued for a subset of orientations that are commensurate with the crystal structure. We support these conclusions with density functional theory calculations that also reveal multivalued surface energies for the set of singular orientations.

• Abstract

In this paper we introduce a new mathematical model for the active contraction of cardiac muscle, featuring different thermo-electric and nonlinear conductivity properties. The passive hyperelastic response of the tissue is described by an orthotropic exponential model, whereas the ionic activity dictates active contraction incorporated through the concept of orthotropic active strain. We use a fully incompressible formulation, and the generated strain modifies directly the conductivity mechanisms in the medium through the pull-back transformation. We also investigate the influence of thermo-electric effects in the onset of multiphysics emergent spatiotemporal dynamics, using nonlinear diffusion. It turns out that these ingredients have a key role in reproducing pathological chaotic dynamics such as ventricular fibrillation during inflammatory events, for instance. The specific structure of the governing equations suggests to cast the problem in mixed-primal form and we write it in terms of Kirchhoff stress, displacements, solid pressure, dimensionless electric potential, activation generation, and ionic variables. We also advance a new mixed-primal finite element method for its numerical approximation, and we use it to explore the properties of the model and to assess the importance of coupling terms, by means of a few computational experiments in 3D.

• Abstract

We propose a numerical method for the simulation of flows from weakly compressible to low Mach regimes in domains with moving boundaries. Non-miscible weakly compressible materials separated by an interface are included as well. The scheme is fully implicit and it exploits the relaxation all-speed scheme introduced in [1]. We consider media with significantly different physical properties and constitutive laws, as fluids and hyperelastic solids. The proposed numerical scheme is fully Eulerian and it is the same for all materials. We present numerical validations by simulating weakly compressible fluid/fluid, solid/solid and solid/fluid interactions.

• Abstract

The efficiency of recently developed gas-kinetic scheme for multimaterial flows is increased through the adoption of a new iteration method in the kinetic non-mixing Riemann solver and an interface sharpening reconstruction method at a cell interface. The iteration method is used to determine the velocity of fluid interface, based on the force balance between both sides due to the incidence and bounce back of particles at the interface. An improved Aitken method is proposed with a simple hybrid of the modified Aitken method (Aitken-Chen) and the Steffensen method. Numerical tests validate its efficiency with significantly less calls to the function not only for the average number but also for the maximum. The new reconstruction is based on the tangent of hyperbola for interface capturing (THINC) but applied only to the volume fraction, which is very simple to be implemented under the stratified framework and capable of resolving fluid interface in mixture. Furthermore, the directional splitting is adopted rather than the previous quasi-one-dimensional method. Typical numerical tests, including several water-gas shock tube flows, and the shock-water cylinder interaction flow show that the improved gas-kinetic scheme can capture fluid interfaces much sharper, while preserving the advantages of the original one.

• Abstract

Stress boundary conditions for the lattice Boltzmann equation that are consistent to Burnett order are proposed and imposed using a moment-based method. The accuracy of the method with complicated spatially-dependent boundary conditions for stress and velocity is investigated using the regularized lid-driven cavity flow. The complete set of boundary conditions, which involve gradients evaluated at the boundaries, are implemented locally. A recently-derived collision operator with modified equilibria and velocity-dependent collision rates to reduce the defect in Galilean invariance is also investigated. Numerical results are in excellent agreement with existing benchmark data and exhibit second-order convergence. The lattice Boltzmann stress field is studied and shown to depart significantly from the Newtonian viscous stress when the ratio of Mach to Reynolds numbers is not negligibly small.

• Abstract

In this paper, we develop the residual based a posteriori error estimates and the corresponding adaptive mesh refinement algorithm for atomistic/continuum (a/c) coupling with finite range interactions in two dimensions. We have systematically derived a new explicitly computable stress tensor formula for finite range interactions. In particular, we use the geometric reconstruction based consistent atomistic/continuum (GRAC) coupling scheme, which is quasi-optimal if the continuum model is discretized by P¹ finite elements. The numerical results of the adaptive mesh refinement algorithm is consistent with the quasi-optimal a priori error estimates.

• Abstract

In this paper, the Riemann solutions of a reduced 6×6 blood flow model in medium-sized to large vessels are constructed. The model is non-strictly hyperbolic and non-conservative in nature, which brings two difficulties of the Riemann problem. One is the appearance of resonance while the other one is loss of uniqueness. The elementary waves include shock wave, rarefaction wave, contact discontinuity and stationary wave. The stationary wave is obtained by solving a steady equation. We construct the Riemann solutions especially when the steady equation has no solution for supersonic initial data. We also verify that the global entropy condition proposed by C.Dafermos can be used here to select the physical relevant solution. The Riemann solutions may contribute to the design of numerical schemes, which can apply to the complex blood flows.

• Abstract

A shifted-inverse iteration is proposed for the finite element discretization of the elastic eigenvalue problem. The method integrates the multigrid scheme and adaptive algorithm to achieve high efficiency and accuracy. Error estimates and optimal convergence for the proposed method are proved. Numerical examples show that the proposed method inherits the advantages of both ingredients and can compute low regularity eigenfunctions effectively.

• Abstract

The peridynamic (PD) theory is a reformulation of the classical theory of continuum solid mechanics and is particularly suitable for the representation of discontinuities in displacement fields and the description of cracks and their evolution in materials, which the classical partial differential equation (PDE) models tend to fail to apply. However, the PD models yield numerical methods with dense stiffness matrices which requires O(N²) memory and O(N³) computational complexity where N is the number of spatial unknowns. Consequently, the PD models are deemed to be computationally very expensive especially for problems in multiple space dimensions. State-based PD models, which were developed lately, can be treated as a great improvement of the previous bond-based PD models. The state-based PD models have more complicated structures than the bond-based PD models. In this paper we develop a fast collocation method for a state-based linear PD model by exploring the structure of the stiffness matrix of the numerical method. The method has an O(N) memory requirement and computational complexity of O(NlogN) per Krylov subspace iteration. Numerical methods are presented to show the utility of the method.

• Abstract

In this paper, we introduce an extension of a splitting method for singularly perturbed equations, the so-called RS-IMEX splitting [Kaiser et al., Journal of Scientific Computing, 70(3), 1390–1407], to deal with the fully compressible Euler equations. The straightforward application of the splitting yields sub-equations that are, due to the occurrence of complex eigenvalues, not hyperbolic. A modification, slightly changing the convective flux, is introduced that overcomes this issue. It is shown that the splitting gives rise to a discretization that respects the low-Mach number limit of the Euler equations; numerical results using finite volume and discontinuous Galerkin schemes show the potential of the discretization.