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2016 | Book

Computational Issues for Optimal Shape Design in Hemodynamics Read chapter Read first chapter

Mathematical Modeling and Optimization of Complex Structures

Editors: Pekka Neittaanmäki, Sergey Repin, Tero Tuovinen

Publisher: Springer International Publishing

Book Series : Computational Methods in Applied Sciences

Part of: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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About this book

This volume contains selected papers in three closely related areas: mathematical modeling in mechanics, numerical analysis, and optimization methods. The papers are based upon talks presented on the International Conference for Mathematical Modeling and Optimization in Mechanics, held in Jyväskylä, Finland, March 6-7, 2014 dedicated to Prof. N. Banichuk on the occasion of his 70th birthday. The articles are written by well-known scientists working in computational mechanics and in optimization of complicated technical models. Also, the volume contains papers discussing the historical development, the state of the art, new ideas, and open problems arising in modern continuum mechanics and applied optimization problems. Several papers are concerned with mathematical problems in numerical analysis, which are also closely related to important mechanical models. The main topics treated include:

* Computer simulation methods in mechanics, physics, and biology;

* Variational problems and methods; minimization algorithms;

* Optimal control problems with distributed and discrete control;

* Shape optimization and shape design problems in science and engineering;

* Sensitivity analysis and parameters optimization of complex systems.

Table of Contents

Frontmatter

Numerical Analysis

Frontmatter
Computational Issues for Optimal Shape Design in Hemodynamics
Abstract
A Fluid-Structure Interaction model is studied for aortic flow, based on Koiter’s shell model for the structure, Navier–Stokes equations for the fluid and transpiration for the coupling. It accounts for wall deformation while yet working on a fixed geometry. The model is established first. Then a numerical approximation is proposed and some tests are given. The model is also used for optimal design of a stent and possible recovery of the arterial wall elastic coefficients by inverse methods.
Olivier Pironneau
Functional A Posteriori Error Estimate for a Nonsymmetric Stationary Diffusion Problem
Abstract
In this paper, a posteriori error estimates of functional type for a stationary diffusion problem with nonsymmetric coefficients are derived. The estimate is guaranteed and does not depend on any particular numerical method. An algorithm for the global minimization of the error estimate with respect to an auxiliary function over some finite dimensional subspace is presented. In numerical tests, global minimization is done over the subspace generated by Raviart-Thomas elements. The improvement of the error bound due to the p-refinement of these spaces is investigated.
Olli Mali
Error Estimates of Uzawa Iteration Method for a Class of Bingham Fluids
Abstract
The paper is concerned with fully guaranteed and computable bounds of errors generated by Uzawa type methods for variational problems in the theory of visco-plastic fluids. The respective estimates have two forms. The first form contains global constants (such as the constant in the Friedrichs inequality for the respective domain), and the second one is based upon decomposition of the domain into a collection of subdomains and uses local constants associated with subdomains.
Marjaana Nokka, Sergey Repin
An Automatic Differentiation Based Approach to the Level Set Method
Abstract
This paper discusses an implementation of the parametric level set method. Adjoint approach is used to perform the sensitivity analysis, but contrary to standard implementations, the state problem is differentiated in its discretized form. The required partial derivatives are computed using tools of automatic differentiation, which avoids the need to derive the adjoint problem from the governing partial differential equation. The augmented Lagrangian approach is used to enforce volume constraints, and a gradient based optimization method is used to solve the subproblems. Applicability of the method is demonstrated by repeating well known compliance minimization studies of a cantilever beam and a Michell type structure. The obtained topologies are in good agreement with reference results.
Jukka I. Toivanen

Mathematical Modeling in Mechanics

Frontmatter
Differential Fluid Mechanics—Harmonization of Analytical, Numerical and Laboratory Models of Flows
Abstract
Concepts of a “solid body motion” and “fluid flow” are compared taking into account the condition of inobservability of a “fluid particle”. General properties of the fundamental set of fluid mechanics equations, accepted for describing fluid flows, are analyzed taking into account the compatibility condition. Hierarchy of periodic flows is classified basing on the order of linearized set of governing equations. Results of theoretical analysis of infinitesimal periodic flows in a stably stratified fluid including periodic internal waves and accompanied family of small scale components are given. Calculations of periodic internal waves propagation and generation in a fluid with arbitrary stable profile of buoyancy are compared with data of schlieren observations in laboratory. Fine flow structure observed behind uniformly towing strip is discussed in context of a given model. Some conclusions and recommendations on improvement techniques of a fluid dynamics experiment are presented.
Yuli D. Chashechkin
Effect of Friction in Sliding Contact of a Sphere Over a Viscoelastic Half-Space
Abstract
Imperfect elasticity of contacting solids results in hysteretic losses during the deformation. In rolling/sliding contact, the losses cause the resistant force, which is called the mechanical component of the friction force. Another cause of the friction is related to the energy losses in formation and breaking of the adhesive bridges between the contacting bodies (adhesive component of friction). In this study the combined effect of the adhesive and mechanical components of friction is analysed based on the consideration of the 3-D contact problem for the spherical indenter sliding with a constant velocity at the boundary of the viscoelastic half-space. The material properties are characterized by the linear viscoelastic solid with one relaxation time. The Coulomb-Amonton law of friction is used to describe the adhesive friction inside the contact region. A numerical-analytical method is developed to solve the contact problem and to find the contact stress distribution. The dependence of the mechanical component of friction force on the adhesive friction coefficient for various load-velocity conditions is studied.
Irina Goryacheva, Fedor Stepanov, Elena Torskaya
Stability of a Tensioned Axially Moving Plate Subjected to Cross-Direction Potential Flow
Abstract
We analyze the stability of an axially moving Kirchhoff plate, subjected to an axial potential flow perpendicular to the direction of motion. The dimensionality of the problem is reduced by considering a cross-directional cross-section of the plate, approximating the axial response with the solution of the corresponding problem of a moving plate in vacuum. The flow component is handled via a Green’s function solution. The stability of the cross-section is investigated via the classical Euler type static linear stability analysis method. The resulting eigenvalue problem is solved numerically using Hermite type finite elements. As a result, the critical velocity and the corresponding eigenfunction are determined. It is seen that even at very low free-stream fluid velocities, the buckling shape may become antisymmetric in the cross direction.
Juha Jeronen, Tytti Saksa, Tero Tuovinen
Multiaxial Fatigue Criteria and Durability of Titanium Compressor Disks in Low- and Very-high-cycle Fatigue Modes
Abstract
Life duration for titanium disks of low temperature part of compressor aero-engine D30-Ku is investigated. Several criteria and models are tested under conditions of low-cycle fatigue (LCF) and very-high-cycle fatigue (VHCF). Parameters of the criteria and models are determined from uniaxial fatigue tests for titanium alloy VT3-1. Stress-strain state of disks and blades is calculated taking into account cyclic centrifugal, aerodynamic, contact loads and blade vibrations. Calculated stresses and strains are used as input data for multiaxial models of LCF and VHCF regimes. Location and scales of fracture as well as time to fracture are calculated. The results of calculations are in good agreement with observations during engine exploitation and correspond to data of fractographic investigations of damaged disks.
Nikolay Burago, Ilia Nikitin
Dynamic Analysis for Axially Moving Viscoelastic Poynting–Thomson Beams
Abstract
This paper is concerned with dynamic characteristics of axially moving beams with the standard linear solid type material viscoelasticity. We consider the Poynting–Thomson version of the standard linear solid model and present the dynamic equations for the axially moving viscoelastic beam assuming that out-of-plane displacements are small. Characteristic behaviour of the beam is investigated by a classical dynamic analysis, i.e., we find the eigenvalues with respect to the beam velocity. With the help of this analysis, we determine the type of instability and detect how the behaviour of the beam changes from stable to unstable.
Tytti Saksa, Juha Jeronen
A Projection Approach to Analysis of Natural Vibrations for Beams with Non-symmetric Cross Sections
Abstract
A projection approach based on the method of integro-differential relations and semi-discretization technique is applied to analyze natural variations of rectilinear elastic beams with non-symmetric cross sections. A numerical algorithm is proposed to compose compatible approximating systems of ordinary differential equations. It is shown that the beam vibrations cannot be separated into four independent types of longitudinal, bending, and torsional motions if a non-symmetric cross section is considered. In this case, all these motions can interact with one another. Nevertheless, only one type of displacement and stress fields makes the largest contribution in the amplitudes of the corresponding vibrations. Several eigenfrequencies and eigenforms of a beam with the isosceles cross section are presented and analyzed.
Vasily Saurin, Georgy Kostin
On Bifurcation Analysis of Implicitly Given Functionals in the Theory of Elastic Stability
Abstract
In this paper, we analyze the stability and bifurcation of elastic systems using a general scheme developed for problems with implicitly given functionals. An asymptotic property for the behaviour of the natural frequency curves in the small vicinity of each bifurcation point is obtained for the considered class of systems. Two examples are given. First is the stability analysis of an axially moving elastic panel, with no external applied tension, performing transverse vibrations. The second is the free vibration problem of a stationary compressed panel. The approach is applicable to a class of problems in mechanics, for example in elasticity, aeroelasticity and axially moving materials (such as paper making or band saw blades).
Nikolay Banichuk, Alexander Barsuk, Juha Jeronen, Pekka Neittaanmäki, Tero Tuovinen

Optimization

Frontmatter
Proximal Bundle Method for Nonsmooth and Nonconvex Multiobjective Optimization
Abstract
We present a proximal bundle method for finding weakly Pareto optimal solutions to constrained nonsmooth programming problems with multiple objectives. The method is a generalization of proximal bundle approach for single objective optimization. The multiple objective functions are treated individually without employing any scalarization. The method is globally convergent and capable of handling several nonconvex locally Lipschitz continuous objective functions subject to nonlinear (possibly nondifferentiable) constraints. Under some generalized convexity assumptions, we prove that the method finds globally weakly Pareto optimal solutions. Concluding, some numerical examples illustrate the properties and applicability of the method.
Marko M. Mäkelä, Napsu Karmitsa, Outi Wilppu
Efficient Parallel Nash Genetic Algorithm for Solving Inverse Problems in Structural Engineering
Abstract
A parallel implementation of a game-theory based Nash Genetic Algorithm (Nash-GAs) is presented in this paper for solving reconstruction inverse problems in structural engineering. We compare it with the standard panmictic genetic algorithm in a HPC environment with up to eight processors. The procedure performance is evaluated on a fifty-five bar sized test case of discrete real cross-section types structural frame. Numerical results obtained on this application show a significant achieved increase of performance using the parallel Nash-GAs approach compared to the standard GAs or Parallel GAs.
Jacques Périaux, David Greiner
Efficient Variational Design Sensitivity Analysis
Abstract
The authors’ variant of variational design sensitivity analysis in structural optimisation is highlighted in detail. A rigorous separation of physical quantities into geometry and displacement mappings based on an intrinsic presentation of continuum mechanics build up the first step. The variations with respect to design and displacements are easily available in a second step. The subsequent discrete matrix expressions are used to formulate the finite element equations in a third step. The fourth step elaborates the derived Matlab implementation while the fifth step shows the computational behaviour for an academic example. Both, the general case of nonlinear structural behaviour and the linearised approximation are outlined. The advocated scheme is compared with the well-known analytical differentiation approach of the discrete finite element equations.
Franz-Joseph Barthold, Nikolai Gerzen, Wojciech Kijanski, Daniel Materna
A Variational Approach to Modelling and Optimization in Elastic Structure Dynamics
Abstract
The paper studies dynamics modelling and control design for elastic systems with distributed parameters. The constitutive laws are specified by an integral equality according with the method of integro-differential relations. The original initial-boundary value problem is reduced to a constrained minimization problem for a nonnegative quadratic functional. A numerical algorithm is developed to solve direct and inverse dynamic problems in linear elasticity based on the Ritz method and finite element technique. The minimized functional is used to define an energy type criteria of solution quality. The efficiency of the approach is demonstrated on the example of a thin rectilinear elastic rod. The control problem is to find motion of a rod from an initial state to a terminal one at a fixed time with the minimal mean energy. The control input is presented by piecewise polynomial displacements on one end of the rod. It is possible to find the exact solution of the problem for a specific relation of the space-time mesh steps. The results of numerical analysis are presented and discussed.
Georgy Kostin, Vasily Saurin
Contact Optimization Problems for Stationary and Sliding Conditions
Abstract
The contact stress distribution is frequently not regular. It may contain singularities reducing the lifetime of machine elements. In order to eliminate such stress singularities, the application of contact pressure control is recommended in the contact conditions. In the paper, several classes of optimization problems are formulated for stationary and sliding contacts. Further, they are illustrated by specific examples. The relation to wear process is made as a natural way to attain the steady state contact profile satisfying the optimality conditions corresponding to minimization of the wear dissipation rate. It is assumed that the displacements and strains are small and the materials of the contacting bodies are elastic.
István Páczelt, Attila Baksa, Zenon Mróz
Some Problems of Multipurpose Optimization for Deformed Bodies and Structures
Abstract
Some problems of multipurpose analysis and optimization of deformed structures and thin-walled structural elements are studied in this paper under some constraints including incomplete data. The first problem is the multipurpose optimization of layered plate made from given set of materials in context of optimization of ballistic limit velocity. Incomplete data concerning the thickness of layers of optimized multilayered shield structure are taken into account. The Pareto-approach and numerical evolutionary method (genetic algorithm) are used for solving of the considered multipurpose problem. The second problem studied in the paper is the shape optimization problem for rigid punch moving on the surface of elastic half-space, which is solved analytically in multipurpose formulation taking into account friction of contacted surfaces, wear of materials and arising pressure distributions. The relative movement is considered in frame of quasi-static formulation. Formulated optimization problem is studied analytically using the developed decomposition approach and exact solutions are obtained for the punch which has a rectangular contact region and moves translationally with a constant velocity.
Alexander Sinitsin, Svetlana Ivanova, Evgeniy Makeev, Nikolay Banichuk
Metadata
Title
Mathematical Modeling and Optimization of Complex Structures
Editors
Pekka Neittaanmäki
Sergey Repin
Tero Tuovinen
Copyright Year
2016
Electronic ISBN
978-3-319-23564-6
Print ISBN
978-3-319-23563-9
DOI
https://doi.org/10.1007/978-3-319-23564-6

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