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)     

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)     

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.     

A comparative study of fiber-matrix interactions by using micromechanical models

The purpose of this study is to describe the interfacial interactions in terms of stress distributions on short fibers in fiber-matrix unit-cell models. The fiber and matrix are subjected to tensile loading. The study consists of three main parts. First, fiber-matrix cell segments are modeled using a 3D finite-element analysis (FEA) with ANSYS. Three different finite-element geometrical unit-cell models are generated in order to simulate the Cox analytical model: a fiber-matrix combination, a single fiber, and a single matrix element. The second part contains the results of 3D FE analyses, which are applied to the Cox formulations by using a computer program developed. In the last part, the analytical solutions for distributions of normal and shear stresses are investigated. Cox 2D linear elasticity solutions, together with finite-element ones, are presented in detail in graphs. The interfacial interactions between the fibers and matrix are also discussed considering the relative changes in the distributions of normal and shear stresses. Russian translation published in Mekhanika Kompozitnykh Materialov, Vol. 44, No. 4, pp. 505–520, July–August, 2008.     

Coupling strategies for the numerical simulation of blood flow in deformable arteries by 3D and 1D models

The fluid structure interaction mechanism in vascular dynamics can be described by either 3D or 1D models, depending on the level of detail of the flow and pressure patterns needed for analysis. A successful strategy that has been proposed in the past years is the so-called geometrical multiscale approach, which consists of coupling both 3D and 1D models so as to use the former only in those regions where details of the fluid flow are needed and describe the remaining part of the vascular network by the simplified 1D model.In this paper we review recently proposed strategies to couple the 3D and 1D models, and within the 3D model, to couple the fluid and structure sub-problems. The 3D/1D coupling strategy relies on the imposition of the continuity of flow rate and total normal stress at the interface. On the other hand, the fluid–structure coupling strategy employs Robin transmission conditions. We present some numerical results and show the effectiveness of the new approaches.     

Solutions of the Klein-Gordon equation on manifolds with variable geometry including dimensional reduction

We develop the recent proposal to use dimensional reduction from the four-dimensional space-time (D = 1 + 3) to the variant with a smaller number of space dimensions D = 1 + d, d < 3, at sufficiently small distances to construct a renormalizable quantum field theory. We study the Klein-Gordon equation with a few toy examples (“educational toys”) of a space-time with a variable spatial geometry including a transition to a dimensional reduction. The examples considered contain a combination of two regions with a simple geometry (two-dimensional cylindrical surfaces with different radii) connected by a transition region. The new technique for transforming the study of solutions of the Klein-Gordon problem on a space with variable geometry into solution of a one-dimensional stationary Schrödinger-type equation with potential generated by this variation is useful. We draw the following conclusions: (1) The signal related to the degree of freedom specific to the higher-dimensional part does not penetrate into the smaller-dimensional part because of an inertial force inevitably arising in the transition region (this is the centrifugal force in our models). (2) The specific spectrum of scalar excitations resembles the spectrum of real particles; it reflects the geometry of the transition region and represents its “fingerprints.” (3) The parity violation due to the asymmetric character of the construction of our models could be related to the CP symmetry violation.     

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)     

Modeling complex multi-component reactive-transport systems: towards a simulation environment based on the concept of a Knowledge Base

A modelling framework within which transport processes in the hydrosphere can be described and interfaced with relevant biogeochemical reactions is presented. Three key elements of this simulation environment are discussed: (1) a numerical engine for solving sets of coupled non-linear process equations; (2) an automated procedure for model code generation (`Automatic Code Generator'); (3) a Web-distributed Knowledge Base (KB) of processes. The Automatic Code Generator translates the information selected in the KB into computer algorithms using the principles defined in the numerical engine. The code CONTRASTE is a first attempt at developing such a modelling framework. It allows one to easily select, adapt and combine a specific set of biogeochemical processes relevant to a user-defined application. The workings of CONTRASTE are described by means of examples which demonstrate how the various components of the simulation environment are coupled and automated. Prospects for future developments towards a fully automated model generation procedure are discussed.     

Markov interest rate models

A general procedure for creating Markovian interest rate models is presented. The models created by this procedure automatically fit within the HJM framework and fit the initial term structure exactly. Therefore they are arbitrage free. Because the models created by this procedure have only one state variable per factor, twoand even three-factor models can be computed efficiently, without resorting to Monte Carlo techniques. This computational efficiency makes calibration of the new models to market prices straightforward. Extended Hull- White, extended CIR, Black-Karasinski, Jamshidian's Brownian path independent models, and Flesaker and Hughston's rational log normal models are one-state variable models which fit naturally within this theoretical framework. The ‘separable’ n-factor models of Cheyette and Li, Ritchken, and Sankarasubramanian - which require n(n + 3)/2 state variables - are degenerate members of the new class of models with n(n + 3)/2 factors. The procedure is used to create a new class of one-factor models, the ‘β-η models.’ These models can match the implied volatility smiles of swaptions and caplets, and thus enable one to eliminate smile error. The β-η models are also exactly solvable in that their transition densities can be written explicitly. For these models accurate - but not exact - formulas are presented for caplet and swaption prices, and it is indicated how these closed form expressions can be used to efficiently calibrate the models to market prices.     

Coupling 3D and 1D fluid-structure interaction models for blood flow simulations

Three-dimensional (3D) simulations of blood flow in medium to large vessels are now a common practice. These models consist of the 3D Navier-Stokes equations for incompressible Newtonian fluids coupled with a model for the vessel wall structure. However, it is still computationally unaffordable to simulate very large sections, let alone the whole, of the human circulatory system with fully 3D fluid-structure interaction models. Thus truncated 3D regions have to be considered. Reduced models, one-dimensional (1D) or zero-dimensional (0D), can be used to approximate the remaining parts of the cardiovascular system at a low computational cost. These models have a lower level of accuracy, since they describe the evolution of averaged quantities, nevertheless they provide useful information which can be fed to the more complex model. More precisely, the 1D models describe the wave propagation nature of blood flow and coupled with the 3D models can act also as absorbing boundary conditions. We consider in this work the coupling of a 3D fluid-structure interaction model with a 1D hyperbolic model. We study the stability of the coupling and present some numerical results. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Efficient approaches for furnace loading of cylindrical parts

This paper addresses the heat treatment operation in a manufacturing plant that produces different types of cylindrical parts. The immediate prior process to heat treatment is furnace-loading, where parts are loaded into baskets. The furnace-loading process is complex and involves issues relating to geometry, and heterogeneity in the parts and in their processing requirements. Currently, furnace-loading is accomplished by operator ingenuity; consequently, the parts loaded in heat treatment often do not use furnace capacity adequately. Efficiency in furnace operation can be achieved by improving basket utilization, which is determined by the furnace-loading process. This paper describes the development of integer and mixed integer LP models for 3D loading of cylindrical parts into furnace baskets. The models consider the exact location of parts to be loaded on the basket and incorporate three models with different objectives; the first addresses the nesting of parts within one another, the second addresses the number of basket layers used, and the third addresses the number of baskets used.     

An armature structure for 3D shapes

We present a novel armature structure for 3D articulated shapes, called SBall short for skeletal balls, which includes two parts： a one-dimensional skeleton and incident balls. Our algorithm mainly focuses on constructing the armature structure. This structure is based on an approximation skeleton which is homotopy equivalent to the shape. Each ball in the structure connects a skeletal joint and an interior region of the shape. The boundary vertices on the shape surface are attached onto the SBall using the power diagram of the ball set. A bilateral O~tering algorithm and a variational segmentation algorithm are proposed to enhance the quality of SBall. Finally, applications of this structure are discussed.     

On the whole spectrum of Timoshenko beams. Part II: further applications

The problem of free vibrations of the Timoshenko beam model has been addressed in the first part of this paper. A careful analysis of the governing equations has shown that the vibration spectrum consists of two parts, separated by a transition frequency, which, depending on the applied boundary conditions, might be itself part of the spectrum. Here, as an extension, the case of a doubly clamped beam is considered. For both parts of the spectrum, the values of natural frequencies are computed and the expressions of eigenmodes are provided: this allows to acknowledge that the nature of vibration modes changes when moving across the transition frequency. This case is a meaningful example of more general ones, where the wave-numbers equation cannot be written in a factorized form and hence must be solved by general root-finding methods for nonlinear transcendental equations. These theoretical results can be used as further benchmarks for assessing the correctness of the numerical values provided by several numerical techniques, e.g. finite element models.     

Proposal for an Invariant Nonlinear Eddy Viscosity Model based on a Consistent 4D Turbulence Modelling Approach

The aim of this new approach is to demonstrate that modelling on a 3D spatial manifold is not equivalent to modelling on a true 4D space-time manifold within Newtonian physics. In the framework of turbulence modelling it will be shown that by geometrically reformulating the averaged Navier-Stokes equations on a 4D non-Riemannian manifold without changing the physical content of the theory, additional modelling restrictions will naturally emerge which are absent in the usual Euclidean (3 + 1)D formulation. By proposing a non-linear eddy viscosity model within the k − ϵ family for high turbulent Reynolds numbers the new invariant modelling approach will demonstrate its clear superiority over current (3 + 1)D modelling techniques. (© 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Dual finite element analysis of three-dimensional axisymmetric elliptic problems. Part I

This paper consists of two parts in which we propose two types of pure dual finite element (FE) models of a three-dimensional (3D) axisymmetric elliptic problem with mixed boundary conditions. Using cylindrical coordinates and weighted Sobolev spaces, a dual 3D problem is transformed into a 2D problem and finite element spaces of divergence-free vector functions are constructed with the help of a stream function. In part I of the paper a priori and a posteriori error estimates are derived for the first type of the FE model. © 1993 John Wiley & Sons, Inc.     

A finite strain viscoplasticitymodel for cold compaction processes ofmetal powders

The production of metal parts by powder metallurgical methods is of increasing interest due to energy and material efficiency of the processes. However, there is still a lack of reliable numerical methods and constitutive models to describe the compaction process from the loose powder to the green (un-sintered) compact via die-compaction or cold isostatic pressing. A new phenomenological constitutive model within the framework of viscoplasticity with internal variables is developed. A central part of the model is a unique and smooth single surface yield function. The applied evolution equations together with the assumed associated flow rule and the convexity of the yield function guarantee thermodynamic consistency of the model. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A Phase Field Model for Ductile to Brittle Failure Mode Transition

The computational modeling of failure mechanisms in solids due to fracture based on sharp crack discontinuities suffers in dynamic problems with complex crack topologies. This can be overcome by a diffusive crack modeling based on the introduction of a crack phase field. We outline a conceptual framework for phase field models of crack propagation in brittle elastic and ductile elastic-plastic solids under dynamic loading and investigate the ductile to brittle failure mode transition observed in the experiment performed by Kalthoff and Winkeler [3]. We develop incremental variational principles and consider their numerical implementations by multi-field finite element methods. To this end, we define energy storage and dissipation functions for the plastic flow including the fracture phase field. The introduction of local history fields that drive the evolution of the crack phase field inspires the construction of robust operator split schemes. (© 2012 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)     

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)