Jiří Nedoma

Domain decomposition for generalized unilateral semi-coercive contact problem with given friction in elasticity

A non-overlapping domain decomposition is applied to a multibody unilateral contact problem with given friction (Trescas model). Approximations are proposed on the basis of the primary variational formulation (in terms of displacements) and linear finite elements. For the discretized problem we employ the concept of local Schur complements, grouping every two subdomains which share a contact area. The proposed algorithm of successive approximations can be recommended for short contacts only, since the contact areas are not divided by interfaces. The numerical examples show the practical efficiency of the algorithm.

3D reconstruction of TMJ after resection of the cyst and the stress-strain analyses

The aim of this article is 3D analysis of the temporomandibular joint (TMJ) patient, who underwent surgery, during which the right TMJ was resected along with the ramus of mandible and consequently the joint was reconstructed with subtotal replacement. The main goal is to give a suitable formulation of mathematical model, which describes the changes of stresses in TMJ incurred after the surgery. The TMJ is a complex, sensitive and highly mobile joint which works bilaterally so each side influences the contralateral joint and because of this the distribution of the stresses is changed in the healthy joint as well. Detailed knowledge about function these are necessary for clinical application of temporomandibular joint prosthesis and also help us estimate the lifetime of the prosthesis a possibilities of alteration in the contra lateral joint components. The geometry for the 3D models is taken from the CT scan date and its numerical solution is based on the theory of semi-coercive unilateral contact problems in linear elasticity. This article provides medical part with case report, discretion of treatment, than the methods of mathematical modeling and his possibilities are described and finally results are reported.

Some recent results on a domain decomposition method in biomechanics of human joints

The contribution deals with mathematical simulations of total joint replacements, simulations of mechanical processes taking place during static loadening and their mathematical description. The nonoverlapping decomposition method for unilateral contact problem and the finite element approach are used. Numerical experiments are presented.

Mathematical and computational methods in biomechanics of human skeletal systems : an introduction

PREFACE. ACKNOWLEDGMENTS. PART I ANATOMY, BIOMECHANICS, AND ALLOARTHROPLASTY OF HUMAN JOINTS. 1 BIOMECHANICS OF THE HUMAN SKELETON ANDTHE PROBLEM OF ALLOARTHROPLASTY. 1.1 Introduction to History of Biomechanics and Alloarthroplasty. 1.2 Biomechanics of Human Joints and Tissues. 2 INTRODUCTION TOTHE ANATOMY OF THE SKELETAL SYSTEM. 2.1 Anatomy of the Skeletal System. 2.2 Human Joints and Their Functions. 2.3 Tribology of Human Joints. 2.4 Biomechanics of the Skeletal System. 3 TOTAL REPLACEMENT OF HUMAN JOINTS. 3.1 View of Arthroplasty Developments. 3.2 Static and Dynamic Loading of Human Joint Replacements. 3.3 Mechanical Destruction of Implants and Demands on Human Joint Arthroplasty. 3.4 Biomaterials in Ostheosynthesis and Alloarthroplasty. 3.5 Artificial Joint Replacements. PART II MATHEMATICAL MODELS OF BIOMECHANICS. 4 BACKGROUND OF BIOMECHANICS. 4.1 Introduction. 4.2 Fundamentals of Continuum Mechanics. 4.3 Background of the Static and Dynamic Continuum Mechanics in Different Rheologies. 4.4 Background of the Quasi-Static and Dynamic Continuum Mechanics in Thermo(visco)elastic Rheology. 5 MATHEMATICAL MODELS OF PARTICULAR PARTS OF THE HUMAN SKELETON AND JOINTS ANDTHEIR REPLACEMENTS BASED ON BOUNDARY VALUE PROBLEM ANALYSES. 5.1 Introduction. 5.2 Mathematical Models of Human Joints and of Their Total Replacements asWell as of Parts of the Human Body. 5.3 Mathematical Models of Human Body Parts and Human Joints and Their Total Replacements Based on the Boundary Value Problems in (Thermo)elasticity. 5.4 Biomechanical Model of a Long Bone. 5.5 Mathematical Model of a Loaded Long Bone Based on Composite Biomaterials. 5.6 Stochastic Approach. 5.7 Mathematical Model of Heat Generation and Heat Propagation in the Neighborhood of the Bone Cement. Problems of Bone Necrosis. 6 MATHEMATICAL ANALYSES AND NUMERICAL SOLUTIONS OF FUNDAMENTAL BIOMECHANICAL PROBLEMS. 6.1 Background of Functional Analysis, Function Spaces, and Variational Inequalities. 6.2 Variational Equations and Inequalities and Their Numerical Approximations. 6.3 Biomechanical Models of Human Joints and Their Total Replacements. 6.4 Stress Strain Analysis of Total Human Joint Replacements in Linear, Nonlinear, Elasticity, and Thermoelasticity: Static Cases, Finite Element Approximations, Homogenization and Domain Decomposition Methods, and Algorithms. 6.5 Stress Strain Analyses of Human Joints and Their Replacements Based on Quasi-Static and Dynamic Multibody Contact Problems in Viscoelastic Rheologies. 6.6 Algorithms. 6.7 Viscoplastic Model of Total Human Joint Replacements. 6.8 Optimal Shape Design in Biomechanics of Human Joint Replacements. 6.9 Worst-Scenario Method in Biomechanics of Human Joint Replacements. 6.10 Biomechanical Models of Human Joint Replacements Coupling Bi- and Unilateral Contacts, Friction, Adhesion, and Wear. PART III BIOMECHANICAL ANALYSES OF PARTICULAR PARTS OF THE HUMAN SKELETON, JOINTS, AND THEIR REPLACEMENTS. 7 BIOMECHANICAL MODELS BASED ON CONTACT PROBLEMS AND BIOMECHANICAL ANALYSES OF SOME HUMAN JOINTS,THEIR TOTAL REPLACEMENTS, AND SOME OTHER PARTS OF THE HUMAN SKELETON. 7.1 Introduction to the Biomechanics of Statically Loaded and of Moving Loaded Human Body. 7.2 Bone Remodeling and the Corresponding Mathematical Model. 7.3 Biomechanical Studies of Cysts, Osteophytes, and of Inter- and Subtrochanteric Osteotomy of the Femur and the Knee Joint. 7.4 Biomechanical Analysis of the Loosened Total Hip Arthroplasty (THA). 7.5 Biomechanical Analysis of the Hip Joint after THA Implanting and Subtrochanteric Osteotomy Healing. 7.6 Analysis of Loaded Tubular Long Bone Filled with Marrow Tissue. 7.7 Numerical Analysis of theWeight-Bearing Total Knee Replacement Analysis of Effect of Axial Angle Changes onWeight-bearing Total Knee Arthroplasty. 7.8 Total Knee Replacement with Rotational Polyethylene Insert. 7.9 Computer-Assisted Surgery in Orthopedics: A Perspective. 7.10 Biomechanical and Mathematical Models of the Thoracolumbal Spine. 7.11 Biomechanical and Mathematical Models of Joints of the Upper Limbs. 7.12 Mathematical and Biomechanical Analyses of the Temporomandibular Joint. APPENDIX. A.1 List of Notations. A.2 Cartesian Tensors. A.3 Some Fundamental Theorems. A.4 Elementary Inequalities. A.5 Finite Element Method. REFERENCES. INDEX.

Modelling of the temporomandibular joints and the role of medical informatics in stomatology

This contribution deals with the 3D analysis of the temporomandibular joint. The main goal is to give the suitable formulation of mathematical model describing with sufficiently accuracy the TMJ and its function. The model is based on the theory of semi-coercive contact problems in linear elasticity, which leads to solving the variational inequality. The numerical solution is based on the finite element approximation of variational inequality corresponding to a generalized semi-coercive contact problem. The obtained numerical results will be discussed. Since the world is globalized new results from the bioinformatics and medical informatics play an important role. Therefore, the problem of an “international global mutual language” for surgeons in stomatology from different countries, and similarly in other branches in medicine, will be shortly discussed.

Temporomandibular joint and its two-dimensional and three-dimensional modelling

Detailed knowledge about the function and morphology of temporomandibular joint are necessary for clinical applications and for analyses of the function of temporomandibular joint prosthesis. Movements of temporomandibular joint are biomechanically sophisticated and are up-to-date not clear. Therefore, the aim of the paper is to give the suitable mathematical approach for analyses of temporomandibular joint. In the paper the analysis of the temporomandibular joint, loaded by traction and compression forces, is presented and shortly discussed. The obtained results, based on two- and three-dimensional mathematical models, represent an introductory work for further studies of biomechanical aspects of temporomandibular joint and of its artificial replacements. The models are based on the theory of contact problems in linear elasticity. For the numerical solutions of the investigated problems the FEM approaches are used and the used algorithm is based on an active-set method for quadratic programming.

On the Stress-Strain Analysis of the Knee Replacement

The paper deals with the stress/strain analysis of an artificial knee joint. Three cases, where femoral part of the knee joint part is cut across under 3, 5 and 7 degrees, are analysed. Finite element method and the nonoverlapping decomposition technique for the contact problem in elasticity are applied. Numerical experiments are presented and discussed.

Numerical modelling of the weight-bearing total knee joint replacement and usage in practice

In the contribution a weight-bearing total knee joint replacement will be based on numerical results on a non-linear contact problem with Coulombian friction in elasticity. The non-overlapping domain decomposition algorithm will be used. The main goal of the contribution represents an application of mathematical modelling and obtained numerical results to the practice-the total knee joint replacement.

Worst scenario and domain decomposition methods in geomechanics

In geomechanics, there are problems whose investigations lead to solving model problems based on variational formulations. Such problems are frequently formulated by variational inequalities as they physically describe the principle of virtual work in its inequality form. In the first part of the contribution, the algorithm for the numerical solution of the discussed variational inequality problem will be investigated. The used parallel algorithm is based on a nonoverlapping domain decomposition method for unilateral contact problem with the given friction and the finite element approach. The conditions of solvability will be presented. In the second part of the contribution, a unilateral contact problem with friction and with uncertain input data in quasi-coupled thermo-elasticity is analysed. Method of worst scenario will be applied to find the most dangerous admissible input data. The solvability of the corresponding worst scenario (antioptimization) problem will be shortly discussed. Numerical experiments, e.g. a tunnel crossing by an active fault will be presented.

Time-Space FE-PDAS Method for Dynamic Unilateral Contact Problem in Viscoelasticity

The paper deals with numerical analysis of a class of multibody dynamic unilateral contact problems. The presented problem describes the seismological model problem, representing a new approach in this branch and of studying propagation of seismic waves through broken up upper parts of the Earth. The semi-implicit finite element and the primal-dual active set strategy methods will be discussed.