Supercloseness of Orthogonal Projections onto Nearby Finite Element Spaces

We derive upper bounds on the difference between the orthogonal projections of a smooth function $u$ onto two finite element spaces that are nearby, in the sense that the support of every shape function belonging to one but not both of the spaces is contained in a common region whose measure tends to zero under mesh refinement. The bounds apply, in particular, to the setting in which the two finite element spaces consist of continuous functions that are elementwise polynomials over shape-regular, quasi-uniform meshes that coincide except on a region of measure $O(h^\gamma)$, where $\gamma$ is a nonnegative scalar and $h$ is the mesh spacing. The projector may be, for example, the orthogonal projector with respect to the $L^2$- or $H^1$-inner product. In these and other circumstances, the bounds are superconvergent under a few mild regularity assumptions. That is, under mesh refinement, the two projections differ in norm by an amount that decays to zero at a faster rate than the amounts by which each projection differs from $u$. We present numerical examples to illustrate these superconvergent estimates and verify the necessity of the regularity assumptions on $u$

Computing stress intensity factors for curvilinear fractures

Computing stress intensity factors around curvilinear fractures from numerical approximations of the elastic fields could be said to be still a largely open problem in computational fracture mechanics. Existing methods fail to provide a convergent rate \u3e0.5, if at all. In this study, I will describe a new formulation of interaction integrals for curvilinear cracks that enable us to compute stress intensity factors around curved fractures with first and second order convergence rates, depending on the accuracy of the approximation used for the elastic fields. We verify the proposed methods through several examples including the benchmark of the circular arc crack problem as well as body force and crack face loaded problems, for which we construct analytical solutions. We further validate and showcase the robustness of the method for the simulation of curvilinear crack propagation [1] where the proposed numerical tools allow for the convergent computation of crack paths in complex fracturing problems. This study is a collaboration with Yongxing Shen, Leon Keer, and Maurizio Chiaramonte. REFERENCE [1] Rangarajan R., Chiaramonte M.M., Shen Y., Hunsweck M.J., Lew A.J. Simulating curvilinear crack propagation with universal meshes. Int. J Numer Meth Engng. (in press)

An Overview of Variational Integrators

The purpose of this paper is to survey some recent advances in variational integrators for both finite dimensional mechanical systems as well as continuum mechanics. These advances include the general development of discrete mechanics, applications to dissipative systems, collisions, spacetime integration algorithms, AVI’s (Asynchronous Variational Integrators), as well as reduction for discrete mechanical systems. To keep the article within the set limits, we will only treat each topic briefly and will not attempt to develop any particular topic in any depth. We hope, nonetheless, that this paper serves as a useful guide to the literature as well as to future directions and open problems in the subject

The search for CDK4/6 inhibitor biomarkers has been hampered by inappropriate proliferation assays

CDK4/6 inhibitors are effective at treating advanced HR+ /HER2- breast cancer, however biomarkers that can predict response are urgently needed. We demonstrate here that previous large-scale screens designed to identify which tumour types or genotypes are most sensitive to CDK4/6 inhibitors have misrepresented the responsive cell lines because of a reliance on metabolic proliferation assays. CDK4/6-inhibited cells arrest in G1 but continue to grow in size, thereby producing more mitochondria. We show that this growth obscures the arrest using ATP-based proliferation assays but not if DNA-based assays are used instead. Furthermore, lymphoma lines, previously identified as the most sensitive, simply appear to respond the best using ATP-based assays because they fail to overgrow during the G1 arrest. Similarly, the CDK4/6 inhibitor abemaciclib appears to inhibit proliferation better than palbociclib because it also restricts cellular overgrowth through off-target effects. DepMap analysis of screening data using reliable assay types, demonstrates that palbociclib-sensitive cell types are also sensitive to Cyclin D1, CDK4 and CDK6 knockout/knockdown, whereas the palbociclib-resistant lines are sensitive to Cyclin E1, CDK2 and SKP2 knockout/knockdown. Potential biomarkers of palbociclib-sensitive cells are increased expression of CCND1 and RB1, and reduced expression of CCNE1 and CDKN2A. Probing DepMap with similar data from metabolic assays fails to reveal these associations. Together, this demonstrates why CDK4/6 inhibitors, and any other anti-cancer drugs that arrest the cell cycle but permit continued cell growth, must now be re-screened against a wide-range of cell types using an appropriate proliferation assay. This would help to better inform clinical trials and to identify much needed biomarkers of response.</p

A constraint-relaxation-recovery mechanism for stomatal dynamics

Models of guard cell dynamics, built on the OnGuard platform, have provided quantitative insights into stomatal function, demonstrating substantial predictive power. However, the kinetics of stomatal opening predicted by OnGuard models were threefold to fivefold slower than observed in vivo. No manipulations of parameters within physiological ranges yielded model kinetics substantially closer to these data, thus highlighting a missing component in model construction. One well‐documented process influencing stomata is the constraining effect of the surrounding epidermal cells on guard cell volume and stomatal aperture. Here, we introduce a mechanism to describe this effect in OnGuard2 constructed around solute release and a decline in turgor of the surrounding cells and its subsequent recovery during stomatal opening. The results show that this constraint–relaxation–recovery mechanism in OnGuard2 yields dynamics that are consistent with experimental observations in wild‐type Arabidopsis, and it predicts the altered opening kinetics of ost2 H+‐ATPase and slac1 Cl− channel mutants. Thus, incorporating solute flux of the surrounding cells implicitly through their constraint on guard cell expansion provides a satisfactory representation of stomatal kinetics, and it predicts a substantial and dynamic role for solute flux across the apoplastic space between the guard cells and surrounding cells in accelerating stomatal kinetics

Importance of Shear in the bcc-to-hcp Transformation in Iron

Iron shows a pressure-induced martensitic phase transformation from the ground state ferromagnetic bcc phase to a nonmagnetic hcp phase at ≈13 GPa. The exact transformation pressure (TP) and pathway are not known. Here we present a multiscale model containing a quantum-mechanics-based multiwell energy function accounting for the bcc and hcp phases of Fe and a construction of kinematically compatible and equilibrated mixed phases. This model suggests that shear stresses have a significant influence on the bcc↔hcp transformation. In particular, the presence of modest shear accounts for the scatter in measured TPs. The formation of mixed phases also provides an explanation for the observed hysteresis in TP