A mixed Finite Element method for 3D elasticity problems at large strains with weakly imposed symmetry

Since the 1970’s, mixed formulations have arisen as an alternative to the classical one-field formulation. In particular, in the realm of physical problems, it appears as a natural solution to solve numerical issues related to incompressibility and localisation phenomena, notably thanks to the introduction of physical variables that are treated as unknowns in the physical equations (in contrast with the classical formulation where typically only one unknown is sought). The finite element methods based on mixed formulations come with ”in-built” a priori error estimators which allows one to control the error in the approximated solution for different fields independently [2]. Another interesting feature of the mixed formulations is that they usually require less regularity for the underlying fields, making those methods applicable to more general problems. However, those methods come with the price of stability. Therefore, compared to classical formulations, extra-carefulness has to be observed in the choice of discretisation spaces. So far, stable mixed finite elements have been proposed for mixed problems in small strain elasticity in 3D. In the context of our work, we show an extension of the mixed finite element for small strain elasticity to large strain problems. In our approach, we use a polar decomposition of the deformation gradient and we approximate simultaneously the rotation and the stretch tensors as two independent fields. Piola Kirchhoff stress and spatial displacements are independent variable fields. We exploit the orthonormality of the rotation tensor using exponential map. From the computational point of view, we use an opensource software, MoFEM [1], developed at the University of Glasgow. MoFEM provides many tools that considerably simplify the analyses like hierarchical shape functions which makes p-refinement effortless or shape functions for (discretised) Hdiv space. Recently, the implementation of the Schur complement procedure significantly helped to improve the efficiency of the code. Numerical examples demonstrate performance of this approach. To improve the stability for 3D mixed problems we introduce a viscosity parameter. The effect of this parameter on load-displacement path is shown and compared with the results obtained by other authors. Further developments allow the inclusion of dissipative phenomena (like plasticity) in the theoretical model using concepts from configurational mechanics

Brittle crack propagation intersecting a contact interface within the framework of Arbitrary Lagrangian-Eulerian description of motion

Configurational mechanics provides the theoretical basis for modelling thermodynamically consistent crack propagation in brittle materials. Using the Arbitrary Lagrangian-Eulerian (ALE) formulation, material kinematics that describe crack surface increment are coupled with spatial kinematics that describe elastic body deformation. To include the contact interaction into the model, a mortar-like formulation was exploited. While the crack surface is distant from the contact interface, contact elements act only in the spatial domain. However, once the crack surface is in the proximity of the contact zone, additional considerations are needed. Firstly, mesh cutting and mesh smoothing due to evolution of the crack surface affect the position of nodes in the material domain, requiring reconstruction of contact elements. Secondly, once the crack front reaches the contact interface, contact pressure provides an additional contribution to configurational forces driving the crack propagation. Therefore, topological changes due to crack propagation and evolution of contact surfaces are strongly coupled. Here we present for the first time the solution of this coupled problem using a monolithic approach. Examples are considered demonstrating the robustness of the proposed framework and evaluating the effect of the contact loading on the crack propagation.</div

Data-driven finite element method

The standard approach to mechanical problems requires the solution to the mathematical equations that describe both the conservation laws and the constitutive relationships, where the latter is obtained after fitting experimental data to a certain material model. In this work, we follow an alternative approach, and develop a Data-Driven (DD) framework for mechanical problems. The conservation laws are satisfied by means of the finite element method; instead of a constitutive relationship we can use experimental data directly, thereby avoiding the need for material model parameters. In this paper, the DD approach is applied to a flow problem in porous media. We present the solution algorithm, demonstrate its efficiency with numerical examples, and, finally, study the influence of the noise present in the data on the results. The framework has been implemented in the open-source finite element software MoFEM

Finite element modelling of bone adaptation of the 3rd metacarpal in racehorses

<p>Fractures of the third metacarpal bone in thoroughbred racehorses are one of the main reasons of euthanasia in the UK. Most fractures occur due to the accumulation of tissue fatigue as a result of repetitive loading [1]. Bone adaptation in response to different loads is known to increase the resistance to fracture. It is believed that finite element analy- sis might provide better understanding of such complicated mechanisms. In this study, a well-established open system thermodynamics approach [2] was adopted to simulate density growth in equine third metacarpal. Implementing the model into a hierarchical approximation framework allowed for the efficient solution of large of problems in 3D, including entire bones with their complicated external structure. Coupled nonlinear gov- erning equations of mass and linear momentum conservation were implemented in the finite element code MOFEM [3]. Its performance was demonstrated with benchmark problems. For example, bone density growth in healthy and prosthetic human femur with a total hip replacement was considered. Solving these problems on a parallel su- percomputer system demonstrated a significant speed-up in computation time due to the application of field splitting. The geometries of the numerical models were accurately represented by processing CT scan images. Furthermore, loading conditions applied to selected bone regions were estimated based on contact forces at mid-stance of a gallop. The obtained density pattern was validated by comparison with the CT data from a race- horse metacarpal bone. It was shown that this method has the potential to accurately model the effect of loading on the bone and could be applied to future studies in order to reduce the risk of injuries. In addition, the prosthetic femur examples presented could be developed into a useful tool to detect possible zones of fracture or resorption, which may lead to loosening of the implants.</p> <ol> <li> <p>[1]  TDH Parkin,PD Clegg, NP French, CJ Proudman,CM Riggs,ER Singer, PM Webbon, and K L Morgan. Risk factors for fatal lateral condylar fracture of the third metacarpus/metatarsus in UK racing. Equine veterinary journal, 37(3):192–199, 2005</p> </li> <li> <p>[2]  E. Kuhl and P. Steinmann. Theory and numerics of geometrically non-linear open system mechanics. International Journal for Numerical Methods in Engineering, 58(11):1593–1615, 2003.</p> </li> <li> <p>[3] Lukasz Kaczmarczyk, Zahur Ullah, Karol Lewandowski, Xuan Meng, Xiao-Yi Zhou, Chris Pearce, and Athanasiadis Ignatios. Mofem-v0.6.6. Nov 2017. http://mofem.eng.gla.ac.uk.</p> </li> </ol> <p> </p

Finite-element modelling of triboelectric nanogenerators accounting for surface roughness

Triboelectric nanogenerators (TENG) transform mechanical energy into electrical energy as a result of contact between suitably chosen surfaces. Surface roughness plays a key role in the performance of contactseparation TENG, defining the ratio of real contact area to the apparent one for a given contact force. We develop a novel numerical approach for coupling mechanical contact and electrostatics equations to simulate TENG behaviour using the finite-element library MoFEM. Good qualitative agreement is found between the numerical predictions of the proposed method, available experimental data and approximate analytical models. The developed finite-element framework can be used for further study of the effect of various TENG parameters on the output performance