Kees Vuik

A mathematical model for the dissolution kinetics of Mg2Si-phases in Al–Mg–Si alloys during homogenisation under industrial conditions

A numerical analysis of the homogenisation treatment of aluminium alloys under industrial circumstances is presented. The basis of this study is a mathematical model which is applicable to the dissolution of stoichiometric multicomponent phases in both finite and infinite ternary media. It handles both complete and incomplete particle dissolution as well as the subsequent homogenisation of the matrix. The precipitate volume fraction and matrix homogeneity are followed during the entire homogenisation treatment. First, the influence of the metallurgical parameters, such as particle size distribution, initial matrix concentration profile and particle geometry on the dissolution- and matrix homogeneity kinetics is analysed. Then, the impact of the heating-rate and local temperature on the homogenisation kinetics is investigated. Conclusions for an optimal homogenisation treatment of aluminium alloys may be drawn. The model presented is general but the calculations were performed for the system Al‐Mg‐Si with an Al-rich matrix and Mg2Si-precipitates.

A numerical method to compute the dissolution of second phases in ternary alloys

Abstract Dissolution of stoichiometric multi-component particles in ternary alloys is an important process occurring during the heat treatment of as-cast aluminium alloys prior to hot extrusion. A mathematical model is proposed to describe such a process. In this model an equation is given to determine the position of the particle interface in time, using two diffusion equations which are coupled by nonlinear boundary conditions at the interface. Some results concerning existence, uniqueness, and monotonicity are given. Furthermore, for an unbounded domain an analytical approximation is derived. The main part of this work is the development of a numerical solution method. Finite differences are used on a grid which changes in time. The discretization of the boundary conditions is important to obtain an accurate solution. The resulting nonlinear algebraic system is solved by the Newton-Raphson method. Numerical experiments illustrate the accuracy of the numerical method. The numerical solution is compared with the analytical approximation.

Lagrangian Formulation of Multiclass Kinematic Wave Model

The kinematic wave model is often used in simulation tools to describe dynamic traffic flow and to estimate and predict traffic states. Discretization of the model is generally based on Eulerian coordinates, which are fixed in space. However, the Lagrangian coordinate system, in which the coordinates move with the velocity of the vehicles, results in more accurate solutions. Furthermore, if the model includes multiple user classes, it describes real traffic more accurately. Such a multiclass model, in contrast to a mixed-class model, treats different types of vehicles (e.g., passenger cars and trucks or vehicles with different origins or destinations, or both) differently. The Lagrangian coordinate system is combined with a multiclass model, and a Lagrangian formulation of the kinematic wave model for multiple user classes is proposed. It is shown that the advantages of the Lagrangian formulation also apply for the multiclass model. Simulations based on the Lagrangian formulation result in more accurate solutions than simulations based on the Eulerian formulation.

Anisotropy in generic multi-class traffic flow models

Traffic flow models and simulation tools are often used for traffic state estimation and prediction. Recently several multi-class models based on the kinematic wave traffic flow model have been introduced. These multi-class models take into account the heterogeneity of both vehicles and drivers. We analyse two important properties of these models: hyperbolicity and anisotropy. Both properties relate to the propagation speed of disturbances, as can be observed in real traffic. We discuss the importance of traffic flow models to be hyperbolic and anisotropic. Moreover, we develop a framework to analyse whether traffic flow models have these properties. Therefore, we derive a generic formulation of multi-class kinematic wave traffic flow models, rewrite it in the Lagrangian formulation and apply eigenvalue analysis to the resulting system of equations. Our analysis shows that most multi-class kinematic wave traffic flow models are indeed hyperbolic and anisotropic under certain modelling conditions.

The dissolution of a stoichiometric second phase in ternary alloys: a numerical analysis

A general numerical model is described for the dissolution kinetics of stoichiometric second phases for one-dimensional cases in ternary systems. The model is applicable to both infinite and finite media and handles both complete and incomplete dissolution. It is shown that the dissolution kinetics of stoichiometric multicomponent second phase particles can differ strongly from that of the mono-element particles. The influence of the soft-impingement, ratio of the diffusion coefficients, stoichiometry, composition and the geometry of the dissolving stoichiometric phase is shown. The model is applied to an AlMgSi-alloy.

A mathematical model for the simulation of the formation and the subsequent regression of hypertrophic scar tissue after dermal wounding

A continuum hypothesis-based model is presented for the simulation of the formation and the subsequent regression of hypertrophic scar tissue after dermal wounding. Solely the dermal layer of the skin is modeled explicitly and it is modeled as a heterogeneous, isotropic and compressible neo-Hookean solid. With respect to the constituents of the dermal layer, the following components are selected as primary model components: fibroblasts, myofibroblasts, a generic signaling molecule and collagen molecules. A good match with respect to the evolution of the thickness of the dermal layer of scars between the outcomes of simulations and clinical measurements on hypertrophic scars at different time points after injury in human subjects is demonstrated. Interestingly, the comparison between the outcomes of the simulations and the clinical measurements demonstrates that a relatively high apoptosis rate of myofibroblasts results in scar tissue that behaves more like normal scar tissue with respect to the evolution of the thickness of the tissue over time, while a relatively low apoptosis rate results in scar tissue that behaves like hypertrophic scar tissue with respect to the evolution of the thickness of the tissue over time. Our ultimate goal is to construct models with which the properties of newly generated tissues that form during wound healing can be predicted with a high degree of certainty. The development of the presented model is considered by us as a step toward their construction.

Superconvergent error estimates for position-dependent smoothness-increasing accuracy-conserving (SIAC) post-processing of discontinuous Galerkin solutions

Superconvergence of discontinuous Galerkin methods is an area of increasing interest due to the ease with which higher order information can be extracted from the approximation. Cockburn, Luskin, Shu, and Suli showed that by applying a B-spline filter to the approximation at the final time, the order of accuracy can be improved from order k+1 to order 2k+1 in the L2-norm for linear hyperbolic equations with periodic boundary conditions (where k is the polynomial degree and h is the mesh element diameter) [Math. Comp. (2003)]. The applicability of this filter for linear hyperbolic problems with non-periodic boundary conditions was computationally extended and renamed a position-dependent smoothness-increasing accuracy-conserving (SIAC) filter by van Slingerland, Ryan, Vuik [SISC (2011)]. However, error estimates in the L2

Branch switching techniques for bifurcation in soil deformation

-norm for this new position-dependent SIAC filter were never given. Furthermore, error estimates in the L-infinity-norm have not been established for the original kernel nor the position-dependent kernel. In this paper, for the first time we establish that it is possible to obtain order s, s=min{2k+1,2k + 2-\frac {d}{2}} accuracy in the L-infinity-norm for the position-dependent SIAC filter, where d is the dimension. Furthermore, we extend the error estimates given by Cockburn et al. so that they are applicable to the entire domain when implementing the position-dependent SIAC filter. We also computationally demonstrate the applicability of this filter for visualization of streamlines.

Fast Newton load flow

The transition from homogeneous to localized deformations during the loading of a soil specimen within a finite element computation is often characterized by a bifurcation point, indicating loss of uniqueness of the solution. The signalling of a bifurcation point is done via the eigenvalues of the structural stiAness matrix resulting from the finite element discretization. Eigenvectors related to negative eigenvalues can be used to perturb a homogeneous state and to obtain a localized deformation mode. This procedure is called branch switching. Several methods are proposed to perform this branch switching. ” 2000 Elsevier Science S.A. All rights reserved.

BI-LANCZOS WITH PARTIAL ORTHOGONALIZATION

The Newton-Raphson method is widely used to solve load flow problems. Traditionally a direct solver is used to solve the linear systems within this method. In this paper we explore the use of an iterative method to solve the linear systems, leading to an inexact Newton-Krylov method. The main parameters of this method are the preconditioner and the forcing terms. Several candidate choices for these parameters are discussed and tested. With the proper preconditioner, and forcing terms, the inexact Newton-Krylov method is shown to greatly improve on using a direct solver.