Variationally consistent methods for Lagrangian dynamics in continuum mechanics

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Aeronautics and Astronautics, 2006.Includes bibliographical references (p. 135-143).Rapid dynamics are commonly encountered in industrial applications such as forging, crash tests and many others. These problems are typically non-linear due to large deformations and/or non-linear constitutive relations. Such problems are typically modelled from a Lagrangian viewpoint, where the mesh is attached to the body; hence, large deformations lead to large distortions in the mesh. Explicit numerical methods are considered to be efficient in these cases where large meshes and small time-steps are employed for spatial and temporal resolution. However, incompressible and nearly incompressible materials pose a problem as the timestep stability restriction in explicit methods becomes increasingly severe. Most of the numerical methods employed for such simulations, are developed from discretization of the equations of motion. Recently, Variational Integrators have been developed where the numerical time integration scheme is developed from a variational principle based on Hamilton's principle of stationary action. Such methods ensure conservation of linear and angular momentum, which lead to more physically consistent simulations.(cont.) In this research, numerical methods addressing incompressibility and mesh distortions have been developed under a variational framework. A variational formulation for mesh adaptation procedures, involving local mesh changes for triangular meshes, is presented. Such procedures are very well suited for explicit methods, without significant expense. Conservation properties of such methods are proved and demonstrated. Further, a Fractional Time-Step method is developed, from a variational framework, for incompressible and nearly incompressible problems. Algorithmic details are presented, followed by examples demonstrating the performance of the method.by Sudeep K. Lahiri.Ph.D

Inverse dynamics of underactuated flexible mechanical systems governed by quasi-linear hyperbolic partial differential equations

Diese Arbeit befasst sich mit der inversen Dynamik unteraktuierter, flexibler, mechanischer Systeme, welche durch quasi-lineare hyperbolische partielle Differentialgleichungen beschrieben werden können. Diese Gleichungnen, sind zeitlich veränderlichen Dirichlet-Randbedingungen unterworfen, welche durch unbekannte, räumlich disjunkte, also nicht kollokierte Neumann-Randbedingungen erzwungen werden. Die zugrundeliegenden Gleichungen werden zunächst abstrakt hergeleitet, bevor verschiedene mechanische Systeme vorgestellt werden können, die mit der eingangs postulierten Formulierung übereinstimmen. Hierzu werden geometrisch exakte Theorien hergeleitet, welche in der Lage sind große Bewegungen schlanker Strukturen wie Seile und Balken, aber auch ganz allgemein, dreidimensionaler Festkörper zu beschreiben. In der Regel werden Anfangs-Randwertprobleme, die in der nichtlinearen Strukturdynamik auftreten, durch Anwendung einer sequentiellen Diskretisierung in Raum und Zeit gelöst. Diese Verfahren basieren für gewöhnlich auf einer räumlichen Diskretisierung mit finiten Elementen, gefolgt von einer geeigneten zeitlichen Diskretisierung, welche meist auf finiten Differenzen beruht. Ein kurzer Überblick über derartige sequentielle Integrationsverfahren für das vorliegende Anfangs-Randwertproblem wird zunächst anhand der direkten Formulierung des Problems gegeben werden. D.h. es wird zunächst das reine Neumann-Randproblem betrachtet, bevor anschließend ganz allgemein, verschiedene Möglichkeiten zur Einbindung etwaiger Dirichlet-Randbedingungen diskutiert werden. Darauf aufbauend wird das Problem der inversen Dynamik im Kontext räumlich diskreter mechanischer Systeme, welche rheonom-holonomen Servo-Bindungen unterliegen, eingeführt. Eine ausführliche Untersuchung dieser Art von gebundenen Systemen soll die grundlegenden Unterschiede zwischen Servo-Bindungen und klassischen Kontakt-Bindungen herausarbeiten. Die daraus resultierenden Folgen für die Entwicklung geeigneter numerisch stabiler Integrationsverfahren können dabei ebenfalls angesprochen werden, bevor zahlreich ausgewählte Beispiele vorgestellt werden können. Aufgrund der sehr eingeschränkten Anwendbarkeit der sequentiellen Lösung der inversen Dynamik in Raum und Zeit, wird eine eingehende Analyse des vorliegenden Anfangs-Randwertproblems unternommen. Vor allem durch die Freilegung der hyperbolischen Struktur der zugrundeliegenden partiellen Differentialgleichungen werden sich weitere Einblicke in das vorliegende Problem erhofft. Die Erforschung der daraus resultierenden Mechanismen der Wellenausbreitung in kontinuierlichen Strukturen öffnet die Tür zur Entwicklung numerisch stabiler Integrationsverfahren für die inverse Dynamik. So kann unter anderem eine Methode vorgestellt werden, die auf der Integration der partiellen Differentialgleichungen entlang charakteristischer Mannigfaltigkeiten beruht. Dies regt zu der Entwicklung neuartiger Galerkinverfahren an, die ebenfalls in dieser Arbeit vorgestellt werden können. Diese neu entwickelten Methoden können anschlie\ss end auf die Steuerung verschiedener mechanischer Systeme angewendet werden. Darüber hinaus können die neuartigen Integrationsverfahren auch auf flexible Mehrkörpersysteme übertragen werden. Angeführt seien hier beispielsweise die kooperative Steuerung eines an mehreren flexiblen Seilen aufgehängten starren Körpers oder die Steuerung des Endeffektors eines flexiblen mehrgliedrigen Schwenkarms. Ausgewählte numerische Beispiele verdeutlichen die Relevanz der hier vorgeschlagenen, in Raum und Zeit simultanen Integration des vorliegenden Anfangs-Randwertproblems

Discrete Differential Geometry of Thin Materials for Computational Mechanics

Instead of applying numerical methods directly to governing equations, another approach to computation is to discretize the geometric structure specific to the problem first, and then compute with the discrete geometry. This structure-respecting discrete-differential-geometric (DDG) approach often leads to new algorithms that more accurately track the physically behavior of the system with less computational effort. Thin objects, such as pieces of cloth, paper, sheet metal, freeform masonry, and steel-glass structures are particularly rich in geometric structure and so are well-suited for DDG. I show how understanding the geometry of time integration and contact leads to new algorithms, with strong correctness guarantees, for simulating thin elastic objects in contact; how the performance of these algorithms can be dramatically improved without harming the geometric structure, and thus the guarantees, of the original formulation; how the geometry of static equilibrium can be used to efficiently solve design problems related to masonry or glass buildings; and how discrete developable surfaces can be used to model thin sheets undergoing isometric deformation

エネルギー関数を持つ発展方程式に対する幾何学的数値計算法

学位の種別: 課程博士審査委員会委員 : (主査)東京大学教授 松尾 宇泰, 東京大学教授 中島 研吾, 東京大学准教授 鈴木 秀幸, 東京大学准教授 長尾 大道, 東京大学准教授 齋藤 宣一University of Tokyo(東京大学

SOLID-SHELL FINITE ELEMENT MODELS FOR EXPLICIT SIMULATIONS OF CRACK PROPAGATION IN THIN STRUCTURES

Crack propagation in thin shell structures due to cutting is conveniently simulated using explicit finite element approaches, in view of the high nonlinearity of the problem. Solidshell elements are usually preferred for the discretization in the presence of complex material behavior and degradation phenomena such as delamination, since they allow for a correct representation of the thickness geometry. However, in solid-shell elements the small thickness leads to a very high maximum eigenfrequency, which imply very small stable time-steps. A new selective mass scaling technique is proposed to increase the time-step size without affecting accuracy. New ”directional” cohesive interface elements are used in conjunction with selective mass scaling to account for the interaction with a sharp blade in cutting processes of thin ductile shells

JST-SPH: a total Lagrangian, stabilised meshless methodology for mixed systems of conservation laws in nonlinear solid dynamics

The combination of linear finite elements space discretisation with Newmark family time-integration schemes has been established as the de-facto standard for numerical analysis of fast solid dynamics. However, this set-up suffers from a series of drawbacks: mesh entanglements and elemental distortion may compromise results of high strain simulations; numerical issues, such as locking and spurious pressure oscillations, are likely to manifest; and stresses usually reach a reduced order of accuracy than velocities. Meshless methods are a relatively new family of discretisation techniques that may offer a solution to problems of excessive distortion experienced by linear finite elements. Amongst these new methodologies, smooth particles hydrodynamics (SPH) is the simplest in concept and the most straightforward to numerically implement. Yet, this simplicity is marred by some shortcomings, namely (i) inconsistencies of the SPH approximation at or near the boundaries of the domain; (ii) spurious hourglass-like modes caused by the rank deficiency associated with nodal integration, and (iii) instabilities arising when sustained internal stresses are predominantly tensile. To deal with the aforementioned SPH-related issues, the following remedies are hereby adopted, respectively: (i) corrections to the kernel functions that are fundamental to SPH interpolation, improving consistency at and near boundaries; (ii) a polyconvex mixed-type system based on a new set of unknown variables (p, F, H and J) is used in place of the displacementbased equation of motion; in this manner, stabilisation techniques from computational fluid dynamics become available; (iii) the analysis is set in a total Lagrangian reference framework. Assuming polyconvex variables as the main unknowns of the set of first order conservation laws helps to establish the existence and uniqueness of analytical solutions. This is a key reassurance for a robust numerical implementation of simulations. The resulting system of hyperbolic first order conservation laws presents analogies to the Euler equations in fluid dynamics. This allows the use of a well-proven stabilisation technique in computational fluid dynamics, the Jameson Schmidt Turkel (JST) algorithm. JST is very effective in damping numerical oscillations, and in capturing discontinuities in the solution that would otherwise be impossible to represent. Finally, we note that the JST-SPH scheme so defined is employed in a battery of numerical tests, selected to check its accuracy, robustness, momentum preservation capabilities, and its viability for solving larger scale, industry-related problems

Numerical Procedures for Nonlinear Static and Dynamic Analyses of Shell Systems of Various Sizes

Tema disertacije so mešani(-hibridni) končni elementi za lupine in integracijske sheme za dinamiko, ki ohranjajo osnovne konstante gibanja. Obravnavani končni elementi temeljijo na dveh geometrijsko točnih teorijah lupin: na modelu z velikimi rotacijami in neraztegljivim smernikom ter modelu brez rotacij z raztegljivim smernikom. Učinkovitost najsodobnejših mešanih(-hibridnih) končnih elementov za lupine je ocenjena na podlagi velikega števila numeričnih primerov. Predlagamo tudi nove, »skoraj optimalne« hibridne-mešane formulacije, ki omogočajo račun dolgih obtežnih korakov, izkazujejo skoraj optimalno konvergenco in so neobčutljive na popačenje mreže. Narejen je pregled implicitnih dinamičnih shem za nelinearno elastodinamiko, ki spadajo med posplošene ? metode in metode, ki ohranjajo (oziroma kontrolirano zmanjšujejo) energijo in ohranjajo gibalno in vrtilno količino. Primerjamo njihove spektralne lastnosti, nagnjenost k močni prekoračitvi analitične rešitve in njihovo natančnost. Z računom niza primerov, kjer rešujemo numerično toge nelinearne enačbe za lupine, ocenimo, kako se te lastnosti prenesejo v nelinearno elastodinamiko. Prikažemo sposobnost obravnavanih shem za kontrolirano disipacijo energije in zmožnost ohranjanja vrtilne količine, s kazalniki napake pa ocenimo njihovo natančnost za nelinearne primere. Izpeljemo sheme, ki ohranjajo/disipirajo energijo ter ohranjajo gibalno in vrtilno količino za hibridno-mešane formulacije lupin. Numerični primeri kažejo, da se robustnost in učinkovitost novih statičnih formulacij prenese tudi v dinamiko. Zaključni del disertacije je povezan z aplikacijo izpeljanih formulacij. Z numerično disipativnimi implicitnimi shemami proučujemo proces uklona lupin. Ocenimo sposobnost teh shem za opis zapletenih procesov uklona, tudi v postkritičnem območju, in pokažemo, da je numerična disipacija višjih frekvenc nujno potrebna za učinkovito dinamično simulacijo teh procesov. Na koncu izpeljane postopke uporabimo še za proučevanje površinskega gubanja ukrivljenih in tankih lupin na mehkih jedrih, vključno s preskoki uklonskih oblik.The topics of the thesis are mixed(-hybrid) finite element formulations for shell-like structures, and implicit time-stepping schemes that preserve basic constants of the motion. The considered finite elements are based on two geometrically exact shell models, in particular, large rotation inextensible-director model and rotation-less extensible-director model. The performance of the current state-of-the-art mixed(-hybrid) shell finite element formulations is assessed by studying a large number of numerical examples. Some novel “near optimal” mixed-hybrid shell finite element formulations are proposed that allow for large solution steps, show near optimal convergence characteristics and display little sensitivity to mesh distortion. As for the non-linear shell elasto-dynamics, we revisit implicit dynamic schemes that belong to the groups of generalized-α methods and energy-conserving/decaying and momentum-conserving methods. We compare their spectral characteristics, the tendency to overshoot and their accuracy. By performing a set of numerical tests for numerically stiff nonlinear shell-like examples, we assess how these features extend to nonlinear elasto-dynamics. We illustrate the ability of the considered schemes to dissipate the energy, to fully or approximately conserve the angular momentum, and we estimate the order of accuracy for nonlinear problems by error indicators. Novel energy-conserving/decaying and momentum-conserving schemes are derived for the previously introduced novel mixed-hybrid shell formulations. The numerical examples demonstrate that the robustness and efficiency of the novel static formulations can be prolonged to dynamics. The final part of the thesis is related to the application of the derived formulations. In particular, the shell buckling process is studied by applying numerically dissipative schemes. The ability of these schemes to handle complex buckling and post-buckling processes is assessed. It is demonstrated that controlled numerical dissipation of higher structural frequencies is absolutely necessary for an efficient simulation of a post-buckling response. Finally, we apply the derived procedures to study the problem of surface wrinkling on curved stiff-shell/soft-core substrates, including the transition between the wrinkling modes