With increasing complexity in structural design, analysts have to make themselves available with methods that can solve any structure at hand. Along with applicability, it is necessary to make analysis codes more accurate and efficient. In the history of modern computational structural analysis, methods such as FEM, BEM, DEM, meshless methods and many others have been developed in order to solve the problem accurately. Most of the methods use some shape functions such as polynomials for the approximation of variables. NURBS is also one of those many shape functions which have been developed not so long ago, but have become popular because of its smoothness and higher continuity. In this regard, IsoGeometric Analysis introduced by Hughes et al. (2005), tries to answer the problem of exactness and efficiency by using the NURBS as basis functions both for geometry and discretization. It is easy to solve any problem in 2D compared to 3D. When the beam is assumed to be in 3D, there are more degrees of freedom, thereby increasing the complexity of kinematic relations. In this project, IsoGeometric analysis has been applied to spatially curved beams, which have all displacements and rotational degrees of freedom, considering St.Venant torsional theory. Similar to FEM, the problems of geometric locking are faced by the NURBS basis functions used in isogeometric analysis as well. Hence, the beam is analysed by comparing different beam formulations and locking removal techniques that have been implemented. Methods to overcome locking, like Hierarchic methods and Hybrid Stress methods, have been studied and codes have been implemented in MATLAB. To make the code applicable to more variety of problems, multi-patch analysis has been implemented. The bending strip method as suggested by Kiendl et al. (2010) and Lagrange multiplier methods have been used for connecting the patches.

The growth and re-orientation of cells can be modelled by continuum-level mathematical models. Since the response to stimuli of cells involves a multitude of biochemical processes, models constitute a set of coupled equations to be solved. In this contribution, we describe a monolithic solution scheme used to solve such coupled systems, which is found to be more robust than the staggered scheme. Using this scheme, we can predict the instantaneous stress fibre reorientation due to different loading and environmental conditions. It has been observed that the response of cells depends on the applied external forces and the substrate properties1. Upon experiencing external mechanical forces, in the presence of cytoplasmic calcium, cells develop stress fibres through the formation of acto-myosin bridge complex, which usually originates and terminates on the cell membrane, at focal adhesions. Stress in stress fibres is modelled through Hill-type equations, in which stress depends non-linearly on the strain rate. The concentration of cytoplasmic calcium which plays a major role in the formation of bridge complex is regulated by the feedback loop coupling the effect of focal adhesion on stress fibre growth and vice-versa in a cyclic loop. Thus, the model involves the mechanical equilibrium of stress fibres coupled with the growth of focal adhesions, in the presence of a feedback loop. We solve this system of non-linear equations simultaneously, by discretising with finite elements and updating through the Newton-Raphson scheme. The Growth of stress fibres is governed by an ODE, which is integrated using a 4th-order Runge-Kutta scheme. The monolithic solution scheme showed that it is stable over a wide range of time steps, whereas the staggered scheme fails. 

Calcification is an important process leading to the formation of bone. In this short project, the role of cyclic stretch on bone formation is studied. Upon adding the bone formation media along with cyclic stretch, calcification is found to increase. Thus, the uni-axial cyclic stretcher with live imaging capability has a wide variety of scientific applications. 

The inner lining of blood vessels, the endothelium, is made up of endothelial cells. Vascular endothelial (VE)-cadherin protein forms a bond with VE-cadherin from neighbouring cells (homophilic bond) to determine the size of gaps between the cells and thereby regulate the size of particles that can cross the endothelium. Chemical cues such as Thrombin, along with mechanical properties of the cell and extracellular matrix (ECM) are known to affect the permeability of endothelial cells. Abnormal permeability is found in patients suffering from diseases including cardiovascular diseases, cancer, and COVID-19. Even though some of the regulatory mechanisms affecting endothelial permeability are well studied, details of how several mechanical and chemical stimuli acting simultaneously affect endothelial permeability are not yet understood. In this article, we present a continuum-level mechanical modelling framework to study the highly dynamic nature of the VE-cadherin bonds. Taking inspiration from the catch slip behaviour that VE-cadherin complexes are known to exhibit, we model VE-cadherin homophilic bond as cohesive contact with damage following a traction-separation law. We explicitly model the actin-cytoskeleton, and substrate to study their role in permeability. Our studies show that mechano-chemical coupling is necessary to simulate the influence of the mechanical properties of the substrate on permeability. Simulations show that shear between cells is responsible for the variation in permeability between bi-cellular and tri-cellular junctions, explaining the phenotypic differences observed in experiments. An increase in the magnitude of traction force that endothelial cells experience results in increased permeability, and it is found that the effect is higher on stiffer ECM. Finally, we show that the cylindrical monolayer exhibits higher permeability than the planar monolayer under unconstrained cases. Thus, we present a contact mechanics-based mechano-chemical model to investigate the variation in permeability of endothelial monolayer due to multiple loads acting simultaneously.