The numerical approximation of partial differential equations (PDEs) posed on complicated geometries, which include a large number of small geometrical features or microstructures, represents a challenging computational problem. Indeed, the use of standard mesh generators, employing simplices or tensor product elements, for example, naturally leads to very fine finite element meshes, and hence the computational effort required to numerically approximate the underlying PDE problem may be prohibitively expensive. As an alternative approach, in this article we present a review of composite/agglomerated discontinuous Galerkin finite element methods (DGFEMs) which employ general polytopic elements. Here, the elements are typically constructed as the union of standard element shapes; in this way, the minimal dimension of the underlying composite finite element space is independent of the number of geometrical features. In particular, we provide an overview of hp-version inverse estimates and approximation results for general polytopic elements, which are sharp with respect to element facet degeneration. On the basis of these results, a priori error bounds for the hp-DGFEM approximation of both second-order elliptic and first-order hyperbolic PDEs will be derived. Finally, we present numerical experiments which highlight the practical application of DGFEMs on meshes consisting of general polytopic elements.
In this article we consider the spatial homogenization of a multiphase model for avascular tumour growth and response to chemotherapeutic treatment. The key contribution of this work is the derivation of a system of homogenized partial differential equations describing macroscopic tumour growth, coupled to transport of drug and nutrient, that explicitly incorporates details of the structure and dynamics of the tumour at the microscale. In order to derive these equations we employ an asymptotic homogenization of a microscopic description under the assumption of strong interphase drag, periodic microstructure, and strong separation of scales. The resulting macroscale model comprises a Darcy flow coupled to a system of reaction-advection partial differential equations. The coupled growth, response, and transport dynamics on the tissue scale are investigated via numerical experiments for simple academic test cases of microstructural information and tissue geometry, in which we observe drug-and nutrient-regulated growth and response consistent with the anticipated dynamics of the macroscale system.
In this work, we consider the spatial homogenization of a coupled transport and fluid–structure interaction model, to the end of deriving a system of effective equations describing the flow, elastic deformation and transport in an active poroelastic medium. The ‘active’ nature of the material results from a morphoelastic response to a chemical stimulant, in which the growth time scale is strongly separated from other elastic time scales. The resulting effective model is broadly relevant to the study of biological tissue growth, geophysical flows (e.g. swelling in coals and clays) and a wide range of industrial applications (e.g. absorbant hygiene products). The key contribution of this work is the derivation of a system of homogenized partial differential equations describing macroscale growth, coupled to transport of solute, that explicitly incorporates details of the structure and dynamics of the microscopic system, and, moreover, admits finite growth and deformation at the pore scale. The resulting macroscale model comprises a Biot-type system, augmented with additional terms pertaining to growth, coupled to an advection–reaction–diffusion equation. The resultant system of effective equations is then compared with other recent models under a selection of appropriate simplifying asymptotic limits.
In this work, we present a pedagogical tumour growth example, in which we apply calibration and validation techniques to an uncertain, Gompertzian model of tumour spheroid growth. The key contribution of this article is the discussion and application of these methods (that are not commonly employed in the field of cancer modelling) in the context of a simple model, whose deterministic analogue is widely known within the community. In the course of the example, we calibrate the model against experimental data that are subject to measurement errors, and then validate the resulting uncertain model predictions. We then analyse the sensitivity of the model predictions to the underlying measurement model. Finally, we propose an elementary learning approach for tuning a threshold parameter in the validation procedure in order to maximize predictive accuracy of our validated model.
Abstract. In this article we consider the a posteriori error estimation and adaptive mesh refinement for the numerical approximation of the travel time functional arising in porous media flows. The key application of this work is in the safety assessment of radioactive waste facilities; in this setting, the travel time functional measures the time taken for a non-sorbing radioactive solute, transported by groundwater, to travel from a potential site deep underground to the biosphere. To ensure the computability of the travel time functional, we employ a mixed formulation of Darcy's law and conservation of mass, together with Raviart-Thomas Hpdiv, Ωq-conforming finite elements. The proposed a posteriori error bound is derived based on a variant of the standard Dual-WeightedResidual approximation, which takes into account the lack of smoothness of the underlying functional of interest. The proposed adaptive refinement strategy is tested on both a simple academic test case and a problem based on the geological units found at the Sellafield site in the UK.Key words. Travel time functional, groundwater flows, adaptivity, goal-oriented a posteriori error estimation, mixed finite element methods 1. Introduction. In recent decades the use of numerical simulations in hydrogeological studies has become commonplace across a range of applications. Amongst these, modelling the post-closure safety performance of deep geological storage of radioactive waste is of particular interest for a posteriori error estimation. Efficient and reliable simulations are required in order to assess the suitability of a specific location for siting a waste repository. Furthermore, there is a critical need to verify any computational results with rigorous error bounds as the effects of an inaccurate simulation could be extremely costly. In the safety assessment of radioactive waste facilities, one of the key quantities of interest is the time taken for a non-sorbing radioactive solute, transported by groundwater, to travel from a potential site deep underground to the biosphere [28,41,44]. Additionally, accurate computation of the travel time has applications in streamline methods for modelling other subsurface flows; for instance in oil and gas reservoir management [36].The suitability of finite element methods has been demonstrated for many of the complex geometries and physical effects that are associated with the numerical approximation of groundwater flow and contaminant transport problems [16,17,20,21,37,41,42]. There are, however, problems associated with the use of nodal-based elements, such as lack of mass conservation at an elemental level and unphysical streamlines, as noted in [21,24,42,43]. These problems are not observed when using a (conforming) mixed finite element method, or finite difference method, in which pressure and velocity are computed simultaneously. Indeed, when employing the latter class of methods on triangular meshes, the tracing of streamlines is straightforward and has been shown to yield physical results, even o...
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