It then explores the potential consequences for health inequalities of the lockdown measures implemented internationally as a response to the COVID-19 pandemic, focusing on the likely unequal impacts of the economic crisis. It then examines how these inequalities in COVID-19 are related to existing inequalities in chronic diseases and the social determinants of health, arguing that we are experiencing a syndemic pandemic. It outlines historical and contemporary evidence of inequalities in pandemics-drawing on international research into the Spanish influenza pandemic of 1918, the H1N1 outbreak of 2009 and the emerging international estimates of socio-economic, ethnic and geographical inequalities in COVID-19 infection and mortality rates. We have conducted a detailed study on the effect of approximate integration for one of the numerical examples shown below.This essay examines the implications of the COVID-19 pandemic for health inequalities. In particular, as the value of the discretization parameter h decreases, the accuracy of the numerical integration should increase proportionally, so that optimal convergence can be obtained. Recent work has shown that integration errors in meshless methods negatively impact the stability of the method when a large number of degrees of freedom is involved. In the other integration cells, we found it is sufficient to use standard Gauss quadrature over a background mesh (such as the Delaunay triangulation of the nodes that takes in to account the discontinuity for the cells cut by the crack).Īn important distinction between meshless methods and standard finite elements is that, in the former, the numerical integration is almost never exact.
#Dx atlas 2.4 crack crack#
This integration method gives excellent results with a low number of integration points and is used on the sub-triangles having the crack tip as a vertex. By looking at the integrands which contain the derivatives of the branch functions, we notice that the Jacobian of the transformation T, will cancel the r − 1 / 2 singularity.
![dx atlas 2.4 crack dx atlas 2.4 crack](https://4.bp.blogspot.com/-ukHD3SpLnuM/WlHBjj_XzXI/AAAAAAAAEiY/OxvFGMK8t74_RoyJGomqi5pHf_MGgf-FwCLcBGAs/s1600/DevilMayCry4_DX9%2B2018-01-07%2B14-31-37-068.png)
Skipped because Gauss points are fixed in an element such that it is unnecessary to recalculate their coordinates Grid nodal velocity and position: v i p k + 1 / 2 = v i p k − 1 / 2 + ∑ I = 1 8 f i I k N I p k m I k Δ t k, x i p k + 1 = x i p k + ∑ I = 1 8 p i I k + 1 / 2 N I p k m I k Δ t k + 1 / 2 Nodal velocity and position: v i I k + 1 / 2 = v i I k − 1 / 2 + Δ t k f i I k / M I, x i I k + 1 = x i I k + Δ t k + 1 / 2 v i I k + 1 / 2 The positions of deformed grid nodes are not required to be calculated. Grid nodal momentum: p i I k + 1 / 2 = p i I k − 1 / 2 + f i I k Δ t k. Nodal force (one-point Gauss quadrature): f i I int, k = − ∑ e N I e, j k σ i j e V e, f i I ext = ∑ e N I e b i e m e Grid nodal force (particle quadrature): f i I int, k = − ∑ p = 1 n p N I p, j k σ i j p m p ρ p, f i I ext, k = ∑ p = 1 n p N I p k b i p k m p Skipped because the mesh nodes carry mass and momentum
![dx atlas 2.4 crack dx atlas 2.4 crack](https://0.academia-photos.com/attachment_thumbnails/66414256/mini_magick20210425-17374-62alr2.png)
Grid nodal mass and momentum: m I k = ∑ p = 1 n p m p N I p k, p i I k − 1 / 2 = ∑ p = 1 n p m p v i p k − 1 / 2 N I p k Thus, the mass matrix in the MPM is no longer a constant matrix as that in the FEM, and has to be recalculated in each time step. Because all the material properties are carried by the particles, the solution on the grid at next time step must be reconstructed from the particle information.
![dx atlas 2.4 crack dx atlas 2.4 crack](https://europepmc.org/articles/PMC6731510/bin/rsif20190218-g3.jpg)
As a result, no fixed mesh connectivity is required in the MPM so that crack propagation could be simulated without changing the mesh connectivity as needed in the FEM. At the end of each time step, the deformed grid could be discarded to employ a new regular grid in the next time step. The computational mesh of a Lagrangian FEM is attached to the material during the whole solution process, while a specific background grid of the MPM is only attached to the material in each time step. Therefore, the constitutive equations are evaluated at Gauss quadrature points in the FEM but at particles in the MPM.Ģ.
![dx atlas 2.4 crack dx atlas 2.4 crack](https://www.unodc.org/images/southeastasiaandpacific/2019/07/tocta/TOCTA-2019-Cover.jpg)
The FEM employs Gauss quadrature to evaluate the integrals in the weak formulation, while the MPM employs particle quadrature. The major differences between the formulations of these two methods are as follows:ġ. Yan Liu, in The Material Point Method, 2017 3.2.4.1 Basic Formulation