arXiv subjects

Gopikrishnan C. Remesan

Publications and source records attributed to Gopikrishnan C. Remesan.

4 recordsLinked to original sources

arXiv2023

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

A unified framework for fourth-order semilinear problems with trilinear nonlinearity and general source 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 Navier-Stokes equations and the von Kármán equations with Morley, discontinuous Galerkin, $C^0$ 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.

math.NA↗

arXiv2022

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

\emph{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 \emph{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.

q-bio.TO↗

arXiv2020

Numerical solution of a two dimensional tumour growth model with moving boundary

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, \cred{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 $\mathbb{P}_2 - \mathbb{P}_1$ Taylor--Hood finite element method for the \cred{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. 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.

math.NA↗

arXiv2019

Numerical solution of the two-phase tumour growth model with moving boundary

A novel numerical technique has been proposed to solve a two-phase tumour growth model in one spatial dimension without needing to account for the boundary dynamics explicitly. The equivalence to the standard definition of a weak solution is proved. The method is tested against equations with analytically known solutions, to illustrate the advantages over the existing techniques. The tumour growth model is solved using the new procedure and showed to be consistent with results available in the literature.

math.NA↗