Equivalent Shell Model of Elastic Gridshells Including the Effect of the Geometric Curvature

In this work, an equivalent continuum of a barrel gridshell is introduced. Constitutive identification procedures based on periodic homogenization are provided in the literature for this purpose, based on a flat Representative Element Volume (REV), notwithstanding that the geometry of the structures concerned is curved. Therefore, the novelty of the present study is the selection of a curved REV to obtain the equivalent elastic constants. The numerical validation of the identification procedure is made comparing gridshell response to that of the equivalent shell under homogeneous load conditions. Finally, in order to highlight the effect of the curved geometry on the constitutive law of the continuum, the response of the proposed model is also compared to that of a continuum obtained from a flat REV.

Three dimensional reconstruction to visualize atrial fibrillation activation patterns on curved atrial geometry

Background The rotational activation created by spiral waves may be a mechanism for atrial fibrillation (AF), yet it is unclear how activation patterns obtained from endocardial baskets are influenced by the 3D geometric curvature of the atrium or ‘unfolding’ into 2D maps. We develop algorithms that can visualize spiral waves and their tip locations on curved atrial geometries. We use these algorithms to quantify differences in AF maps and spiral tip locations between 3D basket reconstructions, projection onto 3D anatomical shells and unfolded 2D surfaces. Methods We tested our algorithms in N = 20 patients in whom AF was recorded from 64-pole baskets (Abbott, CA). Phase maps were generated by non-proprietary software to identify the tips of spiral waves, indicated by phase singularities. The number and density of spiral tips were compared in patient-specific 3D shells constructed from the basket, as well as 3D maps from clinical electroanatomic mapping systems and 2D maps. Results Patients (59.4±12.7 yrs, 60% M) showed 1.7±0.8 phase singularities/patient, in whom ablation terminated AF in 11/20 patients (55%). There was no difference in the location of phase singularities, between 3D curved surfaces and 2D unfolded surfaces, with a median correlation coefficient between phase singularity density maps of 0.985 (0.978–0.990). No significant impact was noted by phase singularities location in more curved regions or relative to the basket location (p>0.1). Conclusions AF maps and phase singularities mapped by endocardial baskets are qualitatively and quantitatively similar whether calculated by 3D phase maps on patient-specific curved atrial geometries or in 2D. Phase maps on patient-specific geometries may be easier to interpret relative to critical structures for ablation planning.

SOME RESULTS ON ∗−RICCI FLOW

In this paper we have introduced the notion of ∗-Ricci flow and shown that ∗-Ricci soliton which was introduced by Kaimakamis and Panagiotidou in 2014 which is a self similar soliton of the ∗-Ricci flow. We have also find the deformation of geometric curvature tensors under ∗-Ricci flow. In the last two section of the paper, we have found the F-functional and ω-functional for ∗-Ricci flow respectively.