Non-regularised inverse finite element analysis for 3D traction force microscopy

The tractions that cells exert on a gel substrate from the observed displacements is an increasingly attractive and valuable information in biomedical experiments. The computation of these tractions requires in general the solution of an inverse problem. Here, we resort to the discretisation with finite elements of the associated direct variational formulation, and solve the inverse analysis using a least square approach. This strategy requires the minimisation of an error functional, which is usually regularised in order to obtain a stable system of equations with a unique solution. In this paper we show that for many common threedimensional geometries, meshes and loading conditions, this regularisation is unnecessary. In these cases, the computational cost of the inverse problem becomes equivalent to a direct finite element problem. For the non-regularised functional, we deduce the necessary and sufficient conditions that the dimensions of the interpolated displacement and traction fields must preserve in order to exactly satisfy or yield a unique solution of the discrete equilibrium equations. We apply the theoretical results to some illustrative examples and to real experimental data. Due to the relevance of the results for biologists and modellers, the article concludes with some practical rules that the finite element discretisation must satisfy.Peer ReviewedPostprint (author's final draft

Sliding joints in 3D beams: conserving algorithms using the master-slave approach

This paper proposes two time-integration algorithms for motion of geometrically exact 3D beams under sliding contact conditions. The algorithms are derived using the socalled master–slave approach, in which constraint equations and the related time-integration of a system of differential and algebraic equations are eliminated by design. Specifically, we study conservation of energy and momenta when the sliding conditions on beams are imposed and discuss their algorithmic viability. Situations where the contact jumps to adjacent finite elements are analysed in detail and the results are tested on two representative numerical examples. It is concluded that an algorithmic preservation of kinematic constraint conditions is of utmost importance.Peer ReviewedPostprint (author's final draft

Porous-based rheological model for tissue fluidisation

It has been experimentally observed that cells exhibit a fluidisation process when subjected to a transient stretch, with an eventual recovery of the mechanical properties upon removal of the applied deformation. This fluidisation process is characterised by a decrease of the storage modulus and an increase of the phase angle. We propose a rheological model which is able to reproduce this combined mechanical response. The model is described in the context of continua and adapted to a cell-centred particle system that simulates cell–cell interactions. Mechanical equilibrium is coupled with two evolution laws: (i) one for the reference configuration, and (ii) another for the porosity or polymer density. The first law depends on the actual strain of the tissue, while the second assumes different remodelling rates during porosity increase and decrease. The theory is implemented on a particle based model and tested on a stretching experiment. The numerical results agree with the experimental measurements for different stretching magnitudes.Peer ReviewedPostprint (author's final draft

Computation of forces from deformed visco-elastic biological tissues

We present a least-squares based inverse analysis of visco-elastic biological tissues. The proposed method computes the set of contractile forces (dipoles) at the cell boundaries that induce the observed and quantified deformations. We show that the computation of these forces requires the regularisation of the problem functional for some load configurations that we study here. The functional measures the error of the dynamic problem being discretised in time with a second-order implicit time-stepping and in space with standard finite elements. We analyse the uniqueness of the inverse problem and estimate the regularisation parameter by means of an L-curved criterion. We apply the methodology to a simple toy problem and to an in vivo set of morphogenetic deformations of the Drosophila embryo.Peer ReviewedPostprint (author's final draft

Computation of bounds for anchor problems in limit analysis and decomposition techniques

Numerical techniques for the computation of strict bounds in limit analyses have been developed for more than thirty years. The efficiency of these techniques have been substantially improved in the last ten years, and have been successfully applied to academic problems, foundations and excavations. We here extend the theoretical background to problems with anchors, interface conditions, and joints. Those extensions are relevant for the analysis of retaining and anchored walls, which we study in this work. The analysis of three-dimensional domains remains as yet very scarce. From the computational standpoint, the memory requirements and CPU time are exceedingly prohibitive when mesh adaptivity is employed. For this reason, we also present here the application of decomposition techniques to the optimisation problem of limit analysis. We discuss the performance of different methodologies adopted in the literature for general optimisation problems, such as primal and dual decomposition, and suggest some strategies that are suitable for the parallelisation of large three-dimensional problems. The propo sed decomposition techniques are tested against representative problems.Peer ReviewedPreprin

AAR-based decomposition algorithm for non-linear convex optimisation

Postprint (published version

AAR-based decomposition method for limit analysis

Postprint (published version

A control Hamiltonian-preserving discretisation for optimal control

© The Author(s), under exclusive licence to Springer Nature B.V. 2023Optimal control theory allows finding the optimal input of a mechanical system modelled as an initial value problem. The resulting minimisation problem may be solved with known direct and indirect methods. We propose time discretisations for both methods, direct midpoint (DMP) and indirect midpoint (IMP) algorithms, which despite their similarities, result in different convergence orders for the adjoint (or co-state) variables. We additionally propose a third time-integration scheme, Indirect Hamiltonian-preserving (IHP) algorithm, which preserves the control Hamiltonian, an integral of the analytical Euler–Lagrange equations of the optimal control problem. We test the resulting algorithms to linear and nonlinear problems with and without dissipative forces: a propelled falling mass subjected to gravity and a drag force, an elastic inverted pendulum, and the locomotion of a worm-like organism on a frictional substrate. To improve the convergence of the solution process of the discretised equations in nonlinear problems, we also propose a computational simple suboptimal initial guess and apply a forward–backward sweep method, which computes each set of variables (state, adjoint and control) in a staggered manner. We demonstrate in our examples their practical advantage for computing optimal solutionsThis work is financially supported by the Spanish Ministry of Science and Innovation under grants CEX2018-000797-S and PID2020-116141GB-I00, and by the Generalitat de Catalunya local government, under grant 2021 SGR 01049.Peer ReviewedPostprint (author's final draft

Conserving time-integration of beams under contact constrains using B-Spline interpolation

The design of energy-momentum algorithms for geometrically exact beams has been achieved more than 15 years ago. However, many of the desired conserivng propeties do not carry over into constrained systems such as beams subjected to sliding contact conditions. We here model such situation and derive a sliding contact conditions that conserves energy and momenta. Basic ingredients of the resulting formulation is the inteprolation of incremental tangent-scaled rotations and a relaxation of the exact sliding condition. We also combine this formulation with a B-Spline interpolation of the beam centroid axis. In this manner, we achieve to smooth the contact loads thrughout the analysis and consequently increase the stability of the numerical model. We demonstrate these advantages and the conserving properties of the algorithm with a set of two-dimensional numerical examples.Peer ReviewedPostprint (published version

Physiology-based model of cell viscoelasticity

The measured viscoelastic properties of biological tissues is the result of the passive and active response of the cells. We propose an evolution law of the remodeling process in the cytoskeleton which is able to mimic the viscous properties of biological cellular tissues. Our model is based on dynamical changes of the resting length. We show that under the small strain regime, the linear rheology models are recovered, with the relaxation time being replaced by the cell resistance to remodel. We implement the one-dimensional model into network systems of two and three dimensions, and show that the same conclusions may be drawn for those systems.Peer ReviewedPostprint (author's final draft