Towards an efficient numerical simulation of complex 3D knee joint motion

We present a time-dependent finite element model of the human knee joint of full 3D geometric complexity together with advanced numerical algorithms needed for its simulation. The model comprises bones, cartilage and the major ligaments, while patella and menisci are still missing. Bones are modeled by linear elastic materials, cartilage by linear viscoelastic materials, and ligaments by one-dimensional nonlinear Cosserat rods. In order to capture the dynamical contact problems correctly, we solve the full PDEs of elasticity with strict contact inequalities. The spatio-temporal discretization follows a time layers approach (first time, then space discretization). For the time discretization of the elastic and viscoelastic parts we use a new contact-stabilized Newmark method, while for the Cosserat rods we choose an energy-momentum method. For the space discretization, we use linear finite elements for the elastic and viscoelastic parts and novel geodesic finite elements for the Cosserat rods. The coupled system is solved by a Dirichlet–Neumann method. The large algebraic systems of the bone–cartilage contact problems are solved efficiently by the truncated non-smooth Newton multigrid method

A dissipative discretization for large deformation frictionless dynamic contact problems

We present a discretization for dynamic large deformation contact problems without friction. Our model is based on Hamilton’s principle, which avoids the explicit appearance of the contact forces. The resulting differential inclusion is discretized in time using a modified midpoint rule. This modification, which concerns the evaluation of the generalized gradient, allows to achieve energy dissipativity. For the space discretization we use a dual-basis mortar method. The resulting spatial algebraic problems are nonconvex minimization problems with nonconvex inequality constraints. These can be solved efficiently using a trust-region SQP framework with a monotone multigrid inner solver

Are Chebyshev-based stability analysis and Urabe's error bound useful features for Harmonic Balance?

Harmonic Balance is one of the most popular methods for computing periodic solutions of nonlinear dynamical systems. In this work, we address two of its major shortcomings: First, we investigate to what extent the computational burden of stability analysis can be reduced by consistent use of Chebyshev polynomials. Second, we address the problem of a rigorous error bound, which, to the authors' knowledge, has been ignored in all engineering applications so far. Here, we rely on Urabe's error bound and, again, use Chebyshev polynomials for the computationally involved operations. We use the error estimate to automatically adjust the harmonic truncation order during numerical continuation, and confront the algorithm with a state-of-the-art adaptive Harmonic Balance implementation. Further, we rigorously prove, for the first time, the existence of some isolated periodic solutions of the forced-damped Duffing oscillator with softening characteristic. We find that the effort for obtaining a rigorous error bound, in its present form, may be too high to be useful for many engineering problems. Based on the results obtained for a sequence of numerical examples, we conclude that Chebyshev-based stability analysis indeed permits a substantial speedup. Like Harmonic Balance itself, however, this method becomes inefficient when an extremely high truncation order is needed as, e.g., in the presence of (sharply regularized) discontinuities.Comment: The final version of this article is available online at https://doi.org/10.1016/j.ymssp.2023.11026

Simplified Vehicle-Bridge Interaction for Medium to Long-span Bridges Subject to Random Traffic Load

This study introduces a simplified model for bridge-vehicle interaction for medium- to long-span bridges subject to random traffic loads. Previous studies have focused on calculating the exact response of the vehicle or the bridge based on an interaction force derived from the compatibility between two systems. This process requires multiple iterations per time step per vehicle until the compatibility is reached. When a network of vehicles is considered, the compatibility equation turns to a system of coupled equations which dramatically increases the complexity of the convergence process. In this study, we simplify the problem into two sub-problems that are decoupled: (a) a bridge subject to random Gaussian excitation, and (b) individual sensing agents that are subject to a linear superposition of the bridge response and the road profile roughness. The study provides sufficient evidence to confirm the simulation approach is valid with a minimal error when the bridge span is medium to long, and the spatio-temporal load pattern can be modeled as random Gaussian. Quantitatively, the proposed approach is over 1,000 times more computationally efficient when compared to the conventional approach for a 500 m long bridge, with response prediction errors below $0.1\%$.Comment: submitted to the Journal of Civil Structural Health Monitorin

3D mixed virtual element formulation for dynamic elasto-plastic analysis

The virtual element method (VEM) for dynamic analyses of nonlinear elasto-plastic problems undergoing large deformations is outlined within this work. VEM has been applied to various problems in engineering, considering elasto-plasticity, multiphysics, damage, elastodynamics, contact- and fracture mechanics. This work focuses on the extension of VEM formulations towards dynamic elasto-plastic applications. Hereby low-order ansatz functions are employed in three dimensions with elements having arbitrary convex or concave polygonal shapes. The formulations presented in this study are based on minimization of potential function for both the static as well as the dynamic behavior. Additionally, to overcome the volumetric locking phenomena due to elastic and plastic incompressibility conditions, a mixed formulation based on a Hu-Washizu functional is adopted. For the implicit time integration scheme, Newmark method is used. To show the model performance, various numerical examples in 3D are presented

Adaptive numerical simulation of contact problems : Resolving local effects at the contact boundary in space and time

This thesis is concerned with the space discretization of static and the space and time discretization of dynamic contact problems. In particular, we derive a new efficient and reliable residual-type a posteriori error estimator for static contact problems and a new space-time connecting discretization scheme for dynamic contact problems in linear elasticity. The methods enable the efficient resolution of local effects at the contact boundary in space and time. Firstly, we prove efficiency and reliability of the new residual-type a posteriori error estimator for the case of simplicial meshes. Several numerical examples in the two- and three-dimensional case show the performance of the residual-type a posteriori error estimator for simplicial and even for non-simplicial meshes. Secondly, for the discretization in time, we present a new method which allows to implicitly compute the local impact times of each node without decreasing the time step size. As it turns out this method gives rise to a generalization of the Newmark scheme which takes into account the local impact times without additional computational effort