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Numerical Methods for Ordinary Differential Equations

Vuik, Cornelis (author), Vermolen, F.J. (author), van Gijzen, M.B. (author), Vuik, Thea (author)

In this book we discuss several numerical methods for solving ordinary differential equations. We emphasize the aspects that play an important role in practical problems. We confine ourselves to ordinary differential equations with the exception of the last chapter in which we discuss the heat equation, a parabolic partial differential equation....

book 2023

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On the fundamental solutions-based inversion of Laplace matrices

Vermolen, F.J. (author), Bakker, D.R. (author), Vuik, Cornelis (author)

The discretisation of the Laplacian results into the well-known Laplace matrix. In the case of a one dimensional problem, an explicit formula for its inverse is derived on the basis of fundamental solutions (Green’s functions) for general boundary conditions. For a linear reaction–diffusion equation, approximations of the inverse are given.

journal article 2022

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A mathematical model for the simulation of the contraction of burns

Koppenol, D.C. (author), Vermolen, F.J. (author), Koppenol-Gonzalez, Gabriela V. (author), Niessen, Frank B. (author), van Zuijlen, Paul P.M. (author), Vuik, Cornelis (author)

A continuum hypothesis-based model is developed for the simulation of the contraction of burns in order to gain new insights into which elements of the healing response might have a substantial influence on this process. Tissue is modeled as a neo-Hookean solid. Furthermore, (myo)fibroblasts, collagen molecules, and a generic signaling molecule...

journal article 2016

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Simulation of Front Instabilities in Density-Driven Flow, Using a Reactive Transport Model for Biogrout Combined with a Randomly Distributed Permeability Field

Van Wijngaarden-van Rossum, W.K. (author), Van Paassen, L.A. (author), Vermolen, F.J. (author), Van Meurs, G.A.M. (author), Vuik, C. (author)

Biogrout is a method to strengthen granular soil, which is based on microbial-induced carbonate precipitation. To model the Biogrout process, a reactive transport model is used. Since high flow rates are undesirable for the Biogrout process, the model equations can be solved with a standard Galerkin finite element method. The Biogrout process...

journal article 2016

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A biomechanical mathematical model for the collagen bundle distribution-dependent contraction and subsequent retraction of healing dermal wounds

Koppenol, D.C. (author), Vermolen, F.J. (author), Niessen, Frank B. (author), van Zuijlen, Paul P.M. (author), Vuik, Cornelis (author)

A continuum hypothesis-based, biomechanical model is presented for the simulation of the collagen bundle distribution-dependent contraction and subsequent retraction of healing dermal wounds that cover a large surface area. Since wound contraction mainly takes place in the dermal layer of the skin, solely a portion of this layer is included...

journal article 2016

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

Koppenol, D.C. (author), Vermolen, F.J. (author), Niessen, Frank B. (author), van Zuijlen, Paul P.M. (author), Vuik, Cornelis (author)

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...

journal article 2016

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A Reactive Transport Model for Biogrout Compared to Experimental Data

Van Wijngaarden, W.K. (author), Van Paassen, L.A. (author), Vermolen, F.J. (author), Van Meurs, G.A.M. (author), Vuik, C. (author)

Biogrout is a method for reinforcement of granular soil. In the Biogrout process, calcium carbonate is produced. This solid connects the grains, and therefore the strength of the soil is increased. The calcium carbonate is formed with the use of micro-organisms. Experiments and numerical simulations have been performed to demonstrate the process...

journal article 2015

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A mathematical model and analytical solution for the fixation of bacteria in biogrout

Van Wijngaarden, W.K. (author), Vermolen, F.J. (author), Van Meurs, G.A.M. (author), Vuik, C. (author)

Biogrout is a new method for soil reinforcement, which is based on microbialinduced carbonate precipitation. Bacteria and reactants are flushed through the soil, resulting in calcium carbonate precipitation and consequent soil reinforcement. Bacteria are crucially important in the Biogrout process since they catalyse the reaction. Hence, to...

journal article 2012

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Stability analysis of the numerical density-enthalpy model

Vermolen, F.J. (author), Vuik, C. (author)

report 2011

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Model based tuning of the release of self healing agents from organic coatings: From Fickian to controlled release kinetics (abstract)

Javierre, E. (author), Garcia Espallargas, S.J. (author), Mol, J.M.C. (author), Vermolen, F.J. (author), Vuik, C. (author), Van der Zwaag, S. (author)

conference paper 2011

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Numerical methods for industrial problems with phase changes

Vermolen, F.J. (author), Vuik, C. (author)

report 2010

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Application of the numerical density-enthalpy method to the multi-phase flow through a porous medium

Ibrahim, D. (author), Vermolen, F.J. (author), Vuik, C. (author)

In this paper we apply a new method to solve multi-phase fluid flow problem for 1D1D and 2D2D fluid systems. This method is developed at TNO and presented in [1] for spatially homogeneous systems. We call this method the numerical density-enthalpy method (or ??-hh method) because density-enthalpy phase diagrams (or ??-hh diagrams) play an...

journal article 2010

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Numerical methods for industrial problems with phase changes

Vermolen, F.J. (author), Vuik, C. (author), Hegen, D. (author)

report 2009

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Modelling Biogrout: A new ground improvement method based on microbial induced carbonate precipitation

Van Wijngaarden, W.K. (author), Vermolen, F.J. (author), Van Meurs, G.A.M. (author), Vuik, C. (author)

report 2009

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A mathematical analysis of physiological and morphological aspects of wound closure

Javierre, E. (author), Vermolen, F.J. (author), Vuik, C. (author), Van der Zwaag, S. (author)

A computational algorithm to study the evolution of complex wound morphologies is developed based on a model of wound closure by cell mitosis and migration due to Adam [Math Comput Model 30(5–6):23–32, 1999]. A detailed analysis of the model provides estimated values for the incubation and healing times. Furthermore, a set of inequalities are...

journal article 2008

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Numerical methods for industrial flow problems

Vuik, C. (author), Vermolen, F.J. (author), Hegen, D. (author)

report 2008

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A mathematical approach to epidermal wound closure: Model analysis and computer simulations

Javierre, E. (author), Vermolen, F.J. (author), Vuik, C. (author), Van der Zwaag, S. (author)

report 2007

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Higher dimensional numerical simulations of precipitate dissolution in multi-component aluminium alloys

Javierre, E. (author), Vuik, C. (author), Vermolen, F.J. (author), Segal, A. (author), Van der Zwaag, S. (author)

In thermal processing of alloys, homogenization of the as-cast microstructure by annealing at such a high temperature that unwanted precipitates are fully dissolved, is required to obtain a microstructure suited to undergo heavy plastic deformation. This process is governed by Fickian diffusion and can be modelled as a Stefan problem. In binary...

conference paper 2006

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A suite of sharp interface models for solid-state transformations in alloys

Vermolen, F.J. (author), Pinson, P. (author), Vuik, C. (author), Van der Zwaag, S. (author)

The work deals in particular with dissolution and growth of particles in solid-state alloys. Several models are presented and numerical solution procedures are summarized briefly. Further, some analytic solutions are considered shortly, which are useful to validate the numerical solutions. One model, which incorporates surface energy effects is...

conference paper 2006

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A level set method for particle dissolution in a binary alloy

Javierre, E. (author), Vuik, C. (author), Vermolen, F.J. (author), Segal, A. (author)

A mathematical model is proposed for the dissolution of stoichiometric particles in binary alloys occurring during the heat treatments of as-cast aluminium alloys prior to hot extrusion. The Level Set method is used to capture the interface location implicitly. The front velocity, only dened on the moving interface, should be advected to the...

report 2005