On the Coupling of 3D-1D Muscle Models for Lumbar Spine Mechanics

The aim of this research is to represent, within one modelling framework, selected parts of the musculoskeletal system using principles of continuum mechanics, while other parts are modelled using lumped-parameter models and principles of Multi-Body Dynamics. The most challenging part within such a framework will be to properly model the transition from 3D to 1D models for skeletal muscles as many of the skeletal muscles extend beyond the selected part. Hence, this paper focuses on an interface condition for the 3D-1D transition within a skeletal muscle. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Coupling 3D and 1D Skeletal Muscle Models

This work introduces a modelling framework towards a forward dynamics simulation of skeletal muscle mechanics that couples three-dimensional (3D) continuum-mechanical-based Finite Element (FE) simulations to rigid body simulations. In this regard, this is a methodological approach, which incorporates different methods to realise simulations of the musculoskeletal system. Such simulations are at present computationally not feasible. To set up such a modelling framework the upper limp is selected. Here, the upper limb consists of an antagonistic muscle pair, the elbow (a simple hinge joint) and an external load. The skeletal muscles are represented by a 3D continuum-mechanical model. The tendons are, for now, assumed to be rigid. The results demonstrate the ability of the system to converge to a physiological realistic position. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Model order reduction of dynamic skeletal muscle models

Forward-dynamics simulations of three-dimensional continuum-mechanical skeletal muscle models are a complex and computationally expensive problem. Considering a fully dynamic modelling framework based on the theory of finite elasticity is challenging as the muscles' mechanical behaviour requires to consider a highly nonlinear, viscoelastic and incompressible material behaviour. The governing equations yield a nonlinear second-order differential algebraic equation (DAE), which represents a challenge to model order reduction (MOR) techniques. This contribution shows the results of the offline phase that could be obtained so far by applying a combination of the proper orthogonal decomposition (POD) and the discrete empirical interpolation method (DEIM). (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Towards modelling skeletal muscle growth and adaptation

Despite an increasing interest in modelling skeletal muscles adaptation, models that address the phenomena within a continuum-mechanical framework using muscle-specific material models are rare in literature. This work focuses on modelling one form of skeletal musle adaptation, namely sarcomerogenesis. Sarcomerogenesis occurs when a given stretch is sustained over a period of time and the number of basic contractile units, which are the sarcomeres, increase. To model sarcomerogenesis within a continuum-mechanical setting, the growth framework based on a multiplicative split of the total deformation gradient is employed. An evolution equation that describes sarcomerogenesis is used and incorporated in a transversally isotropic material model that accounts for a skeletal muscle's active force production capabilities. The material tangent modulus is derived and implemented within the finite-element analysis software. Using this model, one sees that increased number of sarcomeres results in a decreased force response of the muscle tissue over time. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

An atomic interaction-based continuum model for computational multiscale contact mechanics

A computational multiscale contact mechanics model is presented which describes the interaction between deformable solids based on the interaction of individual atoms or molecules. The contact model is formulated in the framework of large deformation continuum mechanics and combines the approaches of molecular modelling [1] and continuum contact mechanics [2]. In the following a brief overview of the contact model is given. Further details can be found in [3], [4] and [5]. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Passive muscle behaviour – experimental and numerical investigations

The material behaviour of skeletal muscles can be decomposed into two parts: an active part, describing the contractile mechanisms, and a passive one, characterising the passive components such as the connective tissue. Computational models are used to support the understanding of complex mechanism inside a muscle. In the present work, we focus on the three-dimensional passive tissue behaviour from the experimental as well as modelling point of view. Therefore, quasi-static experiments have been performed on specimens with regular geometry. By using a three-dimensional optical measurement system the shape of the specimens has been reconstructed at different deformation states. On the modelling side a hyperelastic model with transversal isotropic fibre orientation has been used to describe non-linear stress responses. The model has been validated by performing analyses for different fibre orientations. In summary, it figures out that the proposed modelling approach is able to reflect the experimental results in a satisfying manner. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Full 3D finite element Cosserat formulation with application in layered structures

In this paper, a full three-dimensional (3D) finite element Cosserat formulation is developed within the principles of continuum mechanics in the small deformation framework. The developed finite element formulation is general; however, the proposed constitutive laws incorporate the effect of the internal length parameter of 3D layered continua. The extension of the existing two-dimensional (2D) Cosserat formulation to the 3D framework is novel and is consistent with plate theory which can be considered as the 3D version of beam theory. The results demonstrate a high level of consistency with the analytical solutions predicted by plate theory as well as predictions by alternative numerical techniques such as the discrete element method.     

Electromechanical modelling of skeletal muscle contractions

In the present paper, the aim was to develop a numerical method for optimisation an existing mechanical material model [1] including muscle activation concerning the excitation of skeletal muscles. The modelling idea was a weak and non-monolithic coupling of an electric current expressed by Ohm's law with a hyperelastic muscle model with transversal isotropic characteristics, see [2]. We confirmed the ability of the proposed model by applying on real reconstructed complex muscle geometry. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Bridging scales: An attempt to incorporate cellular responses within a three-dimensional FEM model of active muscle contraction

A mathematical model of the cellular responses of skeletal muscles has been integrated within a three-dimensional biomechanical Finite Element (FEM) model. The FEM model is based on a tri-cubic Hermite Finite Element discretisation of the governing equations of finite elasticity theory and a transversely isotropic constitutive law. To incorporate the cellular information, homogenised values of key physiological parameters, e.g. the pre- and post-power stroke concentration of crossbridge attachments, are computed at the Gauss points of the FEMintegration scheme. These values are then used to modify the stress tensor in such a way that it resembles the contractile response. The advantages of such an improved three-dimensional FEM model are far reaching. These models can be used, for example, to investigate and study local muscle contraction, muscle recruitment patterns, force generation, or fatigue response of skeletal muscles. As an illustrative example, one twitch of the tibialis anterior, in which 25% of the muscle fibres are excited by a nerve stimulus, is simulated. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Three-dimensional reconstruction of M. gastrocnemius contraction

There exist many different approaches investigating the contraction mechanisms of skeletal muscles. Thereby, the mechanical behavior, such as force generation in association with kinematic and microstructure, play an important role in modeling of muscle behavior. Besides the mechanical behaviour, the validation of muscle models requires the geometrical environment, too. The geometry of a muscle can be divided into macrostructure, existing of aponeurosis-tendon-complex (ATC) and muscle tissue (MT), as well as the fascicle architecture, representing the microstructure of the MT. In this study, the macrostructure of the isolated M. gastrocnemius was observed during isometric contraction by using three-dimensional optical measurement systems in combination with mechanical measurement techniques. The surface deformation was reconstructed at specific force and length relationships and further the muscle tissue, aponeurosis, and tendon were distinguished, building up a macroscopic geometrical dataset. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Computer simulations for the mechanical analysis of skeletal muscles

The structure of a skeletal muscle can be seen as a complex hierarchical organisation in which thousands of muscle fibers are arranged within a connective tissue network. Inside of the single muscle fibre many force-producing cells, known as sarcomeres, are connected and take care of the contraction of the whole muscle. The material behaviour of muscles is nonlinear. Due to the fact that muscles can have large deformations in space, geometrical non-linearities must additionally be taken into account. For the simulation of such a behaviour the finite element method is used in the present approach. The material behaviour of the muscle is split into a so-called active and a passive part. To describe the passive part special unit cells consisting of one tetrahedral element and six truss elements have been derived. Additionally to these unit cells other truss elements are attached representing bundles of muscle fibers and therefore the active part of the material behaviour. The contractile behaviour of the muscle is mainly in.uenced by the stretch of the muscle fibres, the shortening velocity and the activation status of the muscle. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Forward dynamics applied to a three-dimensional continuum-mechanical model of the upper limb

This paper introduces, for the first time, a methodology to achieve a forward dynamics simulation of the musculoskeletal system using three-dimensional continuum-mechanical skeletal muscle models. This is achieved by coupling one- and three-dimensional skeletal muscle models. The feasibility of this methodology is demonstrated through a forward dynamics simulation of the upper limb involving the biceps and triceps muscle. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

An integrated mechanistic-neural network modelling for granular systems

A hybrid neural network model is designed to predict the micro-macroscopic characteristics of particulate systems subjected to shearing. The network is initially trained to understand the micro-mechanical characteristics of particulate assemblies, by feeding the results based on three-dimensional discrete element simulations. Given the physical properties of the individual particles and the packing condition of the particulate assemblies under specified loading conditions, the network thus understands the way contact forces are distributed, the orientation of contact (fabric) networks and the evolution of stress tensor during the mechanical loading. These relationships are regarded as soft sensors. Using the signals received from soft sensors, a mechanistic neural network model is constructed to establish the relationship between the micro-macroscopic characteristics of granular assemblies subjected to shearing. The macroscopic results obtained form this hybrid mechanistic neural network modelling for data that were not part of the training signals, is compared with simulations based on discrete element modelling alone and in general, the agreement is good. The hybrid network responds to their inputs at a high speed and can be regarded as a real-time system for understanding the complex behaviour of particulate systems under mechanical process conditions.     

A comparison of static optimization criteria for resolving the 3D muscle redundancy problem during human gait

In this paper, different approaches of static optimization for predicting muscle forces during human walking are investigated. In order to better reflect the true mechanics of the human body, a three-dimensional musculoskeletal model of a single leg is developed. The joint moments generated by muscles during walking are computed from inverse dynamics. The muscle force is estimated by different optimization criteria, each satisfying the moment constraints at all joints and the lower and upper muscle force constraints. Several polynomial and non-polynomial criteria frequently used in literature are studied. Then the results obtained from these calculations are compared with each other. This paper provides an overview of the effects of different optimization criteria on the 3D muscle force distribution problem during human walking. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Numerical Modeling Aspects of Dielectric Elastomer Actuators

In modern actuator technology dielectric elastomers are considered as new materials to realize smart actuators which are known as dielectric elastomer actuators (DEAs). In comparison to piezoceramics actuators, DEAs offer the possibility to achieve large deformations with low actuation forces. This property motivates the implementation as artificial muscles since the deformation-force behavior is similar. Other application fields are pumps, deformable surfaces in aerospace, robotics and haptic feedback. The present work introduces the fundamental concepts to describe the electromechanical coupling in the concept of continuum mechanics for finite deformations. As a benchmark a 3D sandwich actuator setup is taken into account to analyze the mechanical compression stability of the elastomer structure, see [1, 2]. This structure is also considered to study the influence of inhomogeneities in the deformation behavior. For this purpose piezoceramic and air inclusions are considered in the finite element mesh. As a last numerical example an elastomer tube with three pairs of electrodes is simulated numerically to motivate the use of dielectric elastomers as peristaltic pumps. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A finite element method for granular flow through a frictional boundary

A finite element method for the flow of dry granular solids through a domain involving a frictional contact boundary is formulated. The granular material is assumed as a compressible viscous-elastic–plastic continuum. Based on the principles of continuum mechanics, a complete set of equations is developed. The resulting boundary value problem is solved by the finite element method in space and by the finite difference method in time. The derivation of the finite element equations and the mathematical framework of the numerical technique are presented, together with two illustrative examples to demonstrate the validity of the technique.     

A modified approximation of the exponential Cauchy-Born rule for arbitrary shell-like nanostructures

The mechanical behaviour of shell-like nanostructures, such as carbon nanotubes (CNT) can be described within the continuum mechanics by a generalisation of the Cauchy-Born (CB) rule, introduced by Arroyo and Belytschko [1]. However, the finite element approximation presented in the literature requires the reference configuration to be planar. In case the configuration of interest is initially not planar, a preprocessing step is required that isometrically maps the planar reference configuration into the initial configuration. In order to calculate more complex structures, in this paper an extended finite element approximation is presented which is applicable to shell-like atomic structures, whose reference configuration does not need to be planar and consequently the afore-mentioned steps are no longer necessary. The accuracy of the proposed model is verified by two numerical examples. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Extended multiscale FEM for design of beams and frames with complex topology

A numerical method for design of beams and frames with complex topology is proposed. The method is based on extended multi-scale finite element method where beam finite elements are used on coarse scale and continuum elements on fine scale. A procedure for calculation of multi-scale base functions, up-scaling and downscaling techniques is proposed by using a modified version of window method that is used in computational homogenization. Coarse scale finite element is embedded into a frame of a material that is representing surrounding structure in a sense of mechanical properties. Results show that this method can capture displacements, shear deformations and local stress-strain gradients with significantly reduced computational time and memory comparing to full scale continuum model. Moreover, this method includes a special hybrid finite elements for precise modelling of structural joints. Hence, the proposed method has a potential application in large scale 2D and 3D structural analysis of non-standard beams and frames where spatial interaction between structural elements is important.