30 research outputs found

    Periodic steady state response of large scale mechanical models with local nonlinearities

    Get PDF
    AbstractLong term dynamics of a class of mechanical systems is investigated in a computationally efficient way. Due to geometric complexity, each structural component is first discretized by applying the finite element method. Frequently, this leads to models with a quite large number of degrees of freedom. In addition, the composite system may also possess nonlinear properties. The method applied overcomes these difficulties by imposing a multi-level substructuring procedure, based on the sparsity pattern of the stiffness matrix. This is necessary, since the number of the resulting equations of motion can be so high that the classical coordinate reduction methods become inefficient to apply. As a result, the original dimension of the complete system is substantially reduced. Subsequently, this allows the application of numerical methods which are efficient for predicting response of small scale systems. In particular, a systematic method is applied next, leading to direct determination of periodic steady state response of nonlinear models subjected to periodic excitation. An appropriate continuation scheme is also applied, leading to evaluation of complete branches of periodic solutions. In addition, the stability properties of the located motions are also determined. Finally, respresentative sets of numerical results are presented for an internal combustion car engine and a complete city bus model. Where possible, the accuracy and validity of the applied methodology is verified by comparison with results obtained for the original models. Moreover, emphasis is placed in comparing results obtained by employing the nonlinear or the corresponding linearized models

    Multiscale Methods for Random Composite Materials

    Get PDF
    Simulation of material behaviour is not only a vital tool in accelerating product development and increasing design efficiency but also in advancing our fundamental understanding of materials. While homogeneous, isotropic materials are often simple to simulate, advanced, anisotropic materials pose a more sizeable challenge. In simulating entire composite components such as a 25m aircraft wing made by stacking several 0.25mm thick plies, finite element models typically exceed millions or even a billion unknowns. This problem is exacerbated by the inclusion of sub-millimeter manufacturing defects for two reasons. Firstly, a finer resolution is required which makes the problem larger. Secondly, defects introduce randomness. Traditionally, this randomness or uncertainty has been quantified heuristically since commercial codes are largely unsuccessful in solving problems of this size. This thesis develops a rigorous uncertainty quantification (UQ) framework permitted by a state of the art finite element package \texttt{dune-composites}, also developed here, designed for but not limited to composite applications. A key feature of this open-source package is a robust, parallel and scalable preconditioner \texttt{GenEO}, that guarantees constant iteration counts independent of problem size. It boasts near perfect scaling properties in both, a strong and a weak sense on over 15,00015,000 cores. It is numerically verified by solving industrially motivated problems containing upwards of 200 million unknowns. Equipped with the capability of solving expensive models, a novel stochastic framework is developed to quantify variability in part performance arising from localized out-of-plane defects. Theoretical part strength is determined for independent samples drawn from a distribution inferred from B-scans of wrinkles. Supported by literature, the results indicate a strong dependence between maximum misalignment angle and strength knockdown based on which an engineering model is presented to allow rapid estimation of residual strength bypassing expensive simulations. The engineering model itself is built from a large set of simulations of residual strength, each of which is computed using the following two step approach. First, a novel parametric representation of wrinkles is developed where the spread of parameters defines the wrinkle distribution. Second, expensive forward models are only solved for independent wrinkles using \texttt{dune-composites}. Besides scalability the other key feature of \texttt{dune-composites}, the \texttt{GenEO} coarse space, doubles as an excellent multiscale basis which is exploited to build high quality reduced order models that are orders of magnitude smaller. This is important because it enables multiple coarse solves for the cost of one fine solve. In an MCMC framework, where many solves are wasted in arriving at the next independent sample, this is a sought after quality because it greatly increases effective sample size for a fixed computational budget thus providing a route to high-fidelity UQ. This thesis exploits both, new solvers and multiscale methods developed here to design an efficient Bayesian framework to carry out previously intractable (large scale) simulations calibrated by experimental data. These new capabilities provide the basis for future work on modelling random heterogeneous materials while also offering the scope for building virtual test programs including nonlinear analyses, all of which can be implemented within a probabilistic setting

    Méthodes de décomposition de domaine. Application au calcul haute performance

    Get PDF
    This thesis introduces a unified framework for various domain decomposition methods:those with overlap, so-called Schwarz methods, and those based on Schur complements,so-called substructuring methods. It is then possible to switch with a high-level of abstractionbetween methods and to build different preconditioners to accelerate the iterativesolution of large sparse linear systems. Such systems are frequently encountered in industrialor scientific problems after discretization of continuous models. Even though thesepreconditioners naturally exhibit good parallelism properties on distributed architectures,they can prove inadequate numerical performance for complex decompositions or multiscalephysics. This lack of robustness may be alleviated by concurrently solving sparse ordense local generalized eigenvalue problems, thus identifying modes that hinder the convergenceof the underlying iterative methods a priori. Using these modes, it is then possibleto define projection operators based on what is usually referred to as a coarse solver. Theseauxiliary tools tend to solve the aforementioned issues, but typically decrease the parallelefficiency of the preconditioners. In this dissertation, it is shown in three points thatthe newly developed construction is efficient: 1) by performing large-scale numerical experimentson Curie—a European supercomputer, and by comparing it with state of the art2) multigrid and 3) direct solvers.Cette thèse présente une vision unifiée de plusieurs méthodes de décomposition de domaine : celles avec recouvrement, dites de Schwarz, et celles basées sur des compléments de Schur, dites de sous-structuration. Il est ainsi possible de changer de méthodes de manière abstraite et de construire différents préconditionneurs pour accélérer la résolution de grands systèmes linéaires creux par des méthodes itératives. On rencontre régulièrement ce type de systèmes dans des problèmes industriels ou scientifiques après discrétisation de modèles continus. Bien que de tels préconditionneurs exposent naturellement de bonnes propriétés de parallélisme sur les architectures distribuées, ils peuvent s’avérer être peu performants numériquement pour des décompositions complexes ou des problèmes physiques multi-échelles. On peut pallier ces défauts de robustesse en calculant de façon concurrente des problèmes locaux creux ou denses aux valeurs propres généralisées. D’aucuns peuvent alors identifier des modes qui perturbent la convergence des méthodes itératives sous-jacentes a priori. En utilisant ces modes, il est alors possible de définir des opérateurs de projection qui utilisent un problème dit grossier. L’utilisation de ces outils auxiliaires règle généralement les problèmes sus-cités, mais tend à diminuer les performances algorithmiques des préconditionneurs. Dans ce manuscrit, on montre en trois points quela nouvelle construction développée est performante : 1) grâce à des essais numériques à très grande échelle sur Curie—un supercalculateur européen, puis en le comparant à des solveurs de pointe 2) multi-grilles et 3) directs

    A Scalable Two-Level Domain Decomposition Eigensolver for Periodic Schr\"odinger Eigenstates in Anisotropically Expanding Domains

    Full text link
    Accelerating iterative eigenvalue algorithms is often achieved by employing a spectral shifting strategy. Unfortunately, improved shifting typically leads to a smaller eigenvalue for the resulting shifted operator, which in turn results in a high condition number of the underlying solution matrix, posing a major challenge for iterative linear solvers. This paper introduces a two-level domain decomposition preconditioner that addresses this issue for the linear Schr\"odinger eigenvalue problem, even in the presence of a vanishing eigenvalue gap in non-uniform, expanding domains. Since the quasi-optimal shift, which is already available as the solution to a spectral cell problem, is required for the eigenvalue solver, it is logical to also use its associated eigenfunction as a generator to construct a coarse space. We analyze the resulting two-level additive Schwarz preconditioner and obtain a condition number bound that is independent of the domain's anisotropy, despite the need for only one basis function per subdomain for the coarse solver. Several numerical examples are presented to illustrate its flexibility and efficiency.Comment: 30 pages, 7 figures, 2 table

    Lewis Structures Technology, 1988. Volume 1: Structural Dynamics

    Get PDF
    The specific purpose of the symposium was to familiarize the engineering structures community with the depth and range of research performed by the Structures Division of the Lewis Research Center and its academic and industrial partners. Sessions covered vibration control, fracture mechanics, ceramic component reliability, parallel computing, nondestructive testing, dynamical systems, fatigue and damage, wind turbines, hot section technology, structural mechanics codes, computational methods for dynamics, structural optimization, and applications of structural dynamics

    다중 프론탈 해법 기반의 고유치 해석 솔버 병렬 성능 개선 연구

    Get PDF
    학위논문 (박사)-- 서울대학교 대학원 : 기계항공공학부, 2013. 2. 김용협.컴퓨터 그래픽 하드웨어의 발달과 3차원 설계 기술의 진보로 인하여 복잡한 형상의 구조물에 대한 유한요소 모델이 개발되고 있다. 이러한 모델의 수치 해석을 위해서는 많은 계산 비용과 메모리 용량을 요구하게 되었고, 그 결과 병렬 계산이 가능한 고성능의 유한요소 해석 솔버인 IPSAP이 개발되었다. 이 솔버는 응력 해석 모듈, 진동 해석 모듈 등으로 구성되어 있다. 이 중에서 고유 진동수(고유치)와 모드 형상(고유벡터)을 계산하는 진동 해석(모드 해석) 모듈의 경우, 특히 수백개 이상 대량의 고유치를 추출하는 문제의 신속한 해석을 위해서는 많은 메모리 용량을 요구하기 때문에 여전히 다수의 계산 노드들로 구성한 병렬 계산을 요구하고 있다. 따라서 본 연구에서는 분산 메모리 병렬 계산 환경에서의 고유치를 효율적으로 추출하기 위한 방법으로서 계산 성능을 개선하기 위한 연구를 수행하였다. 성능 개선 대상의 고유치 해법은 병렬 계산에 유리한 블록 Lanczos 해법이며, 고성능 직접 선형 해법인 병렬 다중 프론탈 해법이 행렬 분해 과정과 삼각 시스템 연산 과정에서의 효율적이고 안정적인 계산을 위하여 연동되었다. Lanczos 해법을 사용하여 유한요소 구조물의 고유치를 해석하는 과정에서 가장 많은 계산 시간이 요구되는 부분은 유한요소 모델의 자유도 크기와 해석 솔버의 반복 횟수에 따라 차이가 있지만 일반적으로 삼각 시스템 연산에서의 전방 소거 및 후방 대입 과정, 행렬에 대한 Cholesky 분해 과정, 재직교화 과정 순서이다. 본 논문에서는 네트워크로 연결된 계산 노드들간의 병렬 계산 효율성을 높이기 위하여 이러한 고유치 해석 과정에서의 데이터 통신량을 줄이고 응답 대기 시간을 줄이는 것에 초점을 맞추고 연구를 수행하였고, 그 결과 다음과 같은 방법으로 병렬 성능 개선 효과를 얻을 수 있었다. 첫 번째는 Lanczos 벡터에 분산 기법을 적용하여 질량 행렬 곱셈 수행에서의 통신량을 감소시켰고, 두 번째는 행렬 곱셈 라이브러리인 PLASC 내부 서브루틴들 각각의 통신 전송 방법을 설정하는 전송 위상 조합을 최적화하여 원활한 통신 흐름을 가지도록 하였으며, 세 번째는 프론탈 역행렬 계산에 의한 PLASC 서브루틴의 데이터를 독립화하여 LCM 컨셉을 적용 삼각 시스템 연산 과정의 응답 대기 시간을 감소시켰다. 그리고 네 번째는 Cholesky 분해 과정에서 행렬 연산에 관여하는 서브루틴 함수들을 압축하여 통신량을 감소시켰다.제 1 장 서 론 1 제 1.1 절 연구의 배경 및 목적 1 제 1.2 절 연구의 내용 9 제 2 장 고유치 해석 솔버–블록 Lanczos 해법 11 제 2.1 절 병렬 고유치 해석 솔버의 개요 11 제 2.2 절 고유치 해석 솔버의 알고리즘과 분석 19 제 3 장 병렬 계산 성능 27 제 3.1 절 병렬 계산 성능 개선 기법 29 제 3.1.1 절 질량 행렬 곱셈 연산에서의 통신량 감소 30 제 3.1.2 절 삼각 시스템 연산 과정의 응답 대기 시간 감소 33 제 3.1.3 절 Cholesky 분해 과정에서의 통신량 감소 38 제 3.1.4 절 네트워크 전송 위상의 최적화 43 제 3.2 절 개선 기법에 대한 수치 테스트 46 제 4 장 검증, 병렬 성능 비교 및 응용 61 제 4.1 절 고유치 해석 솔버의 검증 61 제 4.2 절 대형 고유치 문제에 대한 병렬 성능 비교 68 제 4.3 절 고유치 해석 솔버의 응용 89 제 5 장 결 론 99 참고문헌 102 Abstract 111Docto

    Parallel processing for nonlinear dynamics simulations of structures including rotating bladed-disk assemblies

    Get PDF
    The principal objective of this research is to develop, test, and implement coarse-grained, parallel-processing strategies for nonlinear dynamic simulations of practical structural problems. There are contributions to four main areas: finite element modeling and analysis of rotational dynamics, numerical algorithms for parallel nonlinear solutions, automatic partitioning techniques to effect load-balancing among processors, and an integrated parallel analysis system

    Government/Industry Workshop on Payload Loads Technology

    Get PDF
    A fully operational space shuttle is discussed which will offer science the opportunity to explore near earth orbit and finally interplanetary space on nearly a limitless basis. This multiplicity of payload/experiment combinations and frequency of launches places many burdens on dynamicists to predict launch and landing environments accurately and efficiently. Two major problems are apparent in the attempt to design for the diverse environments: (1) balancing the design criteria (loads, etc.) between launch and orbit operations, and (2) developing analytical techniques that are reliable, accurate, efficient, and low cost to meet the challenge of multiple launches and payloads. This paper deals with the key issues inherent in these problems, the key trades required, the basic approaches needed, and a summary of the state-of-the-art techniques

    Solution methods for dynamic and non-linear finite element analysis

    Get PDF
    The computer analysis of structures and solids using finite element methods has now taken on very significant proportions. In many cases the safety of a structure may be significantly increased and its cost reduced if an appropriate finite element analysis can be and is performed. In the development and use of finite element methods, we recognize that, considering static linear analysis, already towards the end of the nineteen sixties the methods were highly developed - thus it had taken only about one decade from the inception to the extensive practical use of finite element methods. Although difficulties were still encountered in the linear static analysis of some structures, e.g. complex shells, most of the structures could already be analysed in a routine manner. This situation in engineering analysis was, however, quite different when dynamic or nonlinear conditions had to be considered. Whereas the finite element methods could be developed relatively quickly for linear static analysis, methods for practical dynamic and nonlinear analyses are much more difficult to establish. Although much emphasis has been placed on research in nonlinear analysis, the progress in the development of valuable techniques has been quite slow. The practical objectives in the development of finite element methods for dynamic and nonlinear analysis are, in essence, that we want to be able to analyze increasingly more complex structures which are subjected to loads that vary rapidly - causing dynamic response - and loads of high intensity - causing the structure to respond beyond its linear range. In nonlinear conditions, geometric and/or material nonlinearities may have to be taken into consideration. These analysis conditions are encountered already in many industries (e.g. design of nuclear power plants) and, with the current needs towards usage of new materials and more efficient structures, nonlinear analysis will undoubtedly be required to an increasing extent. Considering research in finite element analysis procedures, emphasis must be placed on the development of reliable, general and cost-effective techniques. The reliability of the analysis techniques is of utmost concern in order that the analyst can employ the methods with confidence. The results of an analysis can only be interpreted with confidence if reliable methods have been employed. The generality and cost-effectiveness of the methods are important in order to produce analysis tools that, in a design office, are applicable to a relatively large number of problems. With the above aims in mind, the development of finite element procedures for dynamic and nonlinear analysis becomes a very formidable task. Not only is it necessary to propose -guided by knowledge and intuition - improved analysis techniques and then to implement and test these methods, but it is of major importance and difficulty to "fully" verify and qualify these theories and their computer program implementations. Whereas the verification and qualification of a finite element method is usually quite straight-forward in linear static analysis, this process may represent the major task in the development of a method for nonlinear analysis. During the last decade I have endeavored to advance the state-of-the-art of general and reliable finite element analysis procedures for dynamic and nonlinear response calculations

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

    Get PDF
    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
    corecore