Indian Institute of Technology Palakkad&nbsp;(IIT Palakkad or IIT PKD) is a public autonomous&nbsp;engineering&nbsp;and&nbsp;research&nbsp;institute located in&nbsp;Palakkad,&nbsp;Kerala. It was&nbsp;one of the five new&nbsp;IITs&nbsp;proposed in the&nbsp;2014 Union budget of India. The campus was inaugurated on 3 August 2015. The institute has 94 faculty, 978 students, and 63 non-teaching staffs.

IT Palakkad currently offers four-year Bachelor of Technology (B.Tech) and two years MS programmes in 8 different engineering disciplines including&nbsp;Computer Science, Electrical, Mechanical, Civil among others.

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IIT Palakkad

A unified framework for fourth-order semilinear problems with trilinear nonlinearity and general sources allows for quasi-best approximation with lowest-order finite element methods. This paper establishes the stability and a priori error control in the piecewise energy and weaker Sobolev norms under minimal hypotheses. Applications include the stream function vorticity formulation of the incompressible 2D NavierStokes equations and the von Kármán equations with Morley, discontinuous Galerkin, C0 interior penalty, and weakly over-penalized symmetric interior penalty schemes. The proposed new discretizations consider quasi-optimal smoothers for the source term and smoother-type modifications inside the nonlinear terms.

Department of Mathematics

Fulltext

Numerische Mathematik

Unified a priori analysis of four second-order FEM for fourth-order quadratic semilinear problems

In vitro experiments in which tumour cells are seeded in a gelatinous medium, or hydrogel, show how mechanical interactions between tumour cells and the tissue in which they are embedded, together with local levels of an externally-supplied, diffusible nutrient (e.g., oxygen), affect the tumour’s growth dynamics. In this article, we present a mathematical model that describes these in vitro experiments. We use the model to understand how tumour growth generates mechanical deformations in the hydrogel and how these deformations in turn influence the tumour’s growth. The hydrogel is viewed as a nonlinear hyperelastic material and the tumour is modelled as a two-phase mixture, comprising a viscous tumour cell phase and an isotropic, inviscid interstitial fluid phase. Using a combination of numerical and analytical techniques, we show how the tumour’s growth dynamics change as the mechanical properties of the hydrogel vary. When the hydrogel is soft, nutrient availability dominates the dynamics: the tumour evolves to a large equilibrium configuration where the proliferation rate of nutrient-rich cells on the tumour boundary balances the death rate of nutrient-starved cells in the central, necrotic core. As the hydrogel stiffness increases, mechanical resistance to growth increases and the tumour’s equilibrium size decreases. Indeed, for small tumours embedded in stiff hydrogels, the inhibitory force experienced by the tumour cells may be so large that the tumour is eliminated. Analysis of the model identifies parameter regimes in which the presence of the hydrogel drives tumour elimination. © 2022, The Author(s), under exclusive licence to Springer-Verlag GmbH Germany, part of Springer Nature.

World University root node

Journal of Mathematical Biology

Two-phase model of compressive stress induced on a surrounding hyperelastic medium by an expanding tumour

A one-spatial dimensional tumour growth model Breward et al. (2001, 2002, 2003) that consists of three dependent variables of space and time is considered. These variables are volume fraction of tumour cells, velocity of tumour cells, and nutrient concentration. The model variables satisfy a coupled system of semilinear advection equation (hyperbolic), simplified linear Stokes equation (elliptic), and semilinear diffusion equation (parabolic) with appropriate conditions on the time-dependent boundary, which is governed by an ordinary differential equation. A reformulation of the model defined in a larger, fixed time-space domain is employed to overcome theoretical difficulties related to the time-dependent boundary. This reformulation reduces the complexity of the model by removing the need to explicitly track the time-dependent boundary. A numerical scheme that employs a finite volume method for the hyperbolic equation, a finite element method for the elliptic equation, and a backward Euler in time-mass lumped finite element in space method for the parabolic equation is developed. We establish the existence of a time interval $(0,T_{\ast })$ over which a convergent subsequence of the numerical approximations can be extracted using compactness techniques. The limit of any such convergent subsequence is proved to be a weak solution of the continuous model in an appropriate sense, which is called a threshold solution. Numerical tests and justifications that support the theoretical findings conclude the paper. © 2021 The Author(s) 2021. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved.

IMA Journal of Numerical Analysis

Convergence analysis of a numerical scheme for a tumour growth model

A uniform bounded variation estimate for finite volume approximations of the nonlinear scalar conservation law ℓ tα + div(uf(α)) = 0 in two and three spatial dimensions with an initial data of bounded variation is established. We assume that the divergence of the velocity div(u) is of bounded variation instead of the classical assumption that div(u) is zero. The finite volume schemes analysed in this article are set on nonuniform Cartesian grids. A uniform bounded variation estimate for finite volume solutions of the conservation law ℓ tα + div(F(t,x,α)) = 0, where divxF ≠ 0 on nonuniform Cartesian grids is also proved. Such an estimate provides compactness for finite volume approximations in Lp spaces, which is essential to prove the existence of a solution for a partial differential equation with nonlinear terms in α, when the uniqueness of the solution is not available. This application is demonstrated by establishing the existence of a weak solution for a model that describes the evolution of initial stages of breast cancer proposed by Franks et al. [J. Math. Biol. 47 (2003) 424-452]. The model consists of four coupled variables: tumour cell concentration, tumour cell velocity-pressure, and nutrient concentration, which are governed by a hyperbolic conservation law, viscous Stokes system, and Poisson equation, respectively. Results from numerical tests are provided and they complement theoretical findings. © EDP Sciences, SMAI 2021.

ESAIM: Mathematical Modelling and Numerical Analysis

Strong bounded variation estimates for the multi-dimensional finite volume approximation of scalar conservation laws and application to a tumour growth model

We consider a biphasic continuum model for avascular tumour growth in two spatial dimensions, in which a cell phase and a fluid phase follow conservation of mass and momentum. A limiting nutrient that follows a diffusion process controls the birth and death rate of the tumour cells. The cell volume fraction, cell velocity–fluid pressure system, and nutrient concentration are the model variables. A coupled system of a hyperbolic conservation law, a viscous fluid model, and a parabolic diffusion equation governs the dynamics of the model variables. The tumour boundary moves with the normal velocity of the outermost layer of cells, and this time-dependence is a challenge in designing and implementing a stable and fast numerical scheme. We recast the model into a form where the hyperbolic equation is defined on a fixed extended domain and retrieve the tumour boundary as the interface at which the cell volume fraction decreases below a threshold value. This procedure eliminates the need to track the tumour boundary explicitly and the computationally expensive re-meshing of the time-dependent domains. A numerical scheme based on finite volume methods for the hyperbolic conservation law, Lagrange P2- P1 Taylor-Hood finite element method for the viscous system, and mass-lumped finite element method for the parabolic equations is implemented in two spatial dimensions, and several cases are studied. We demonstrate the versatility of the numerical scheme in catering for irregular and asymmetric initial tumour geometries. When the nutrient diffusion equation is defined only in the tumour region, the model depicts growth in free suspension. On the contrary, when the nutrient diffusion equation is defined in a larger fixed domain, the model depicts tumour growth in a polymeric gel. We present numerical simulations for both cases and the results are consistent with theoretical and heuristic expectations such as early linear growth rate and preservation of radial symmetry when the boundary conditions are symmetric. The work presented here could be extended to include the effect of drug treatment of growing tumours. © 2020, Springer Science+Business Media, LLC, part of Springer Nature.