Computational Engineering

The focus of this Computational Engineering Workshop was on the mathematical foundation of state-of-the-art and emerging finite element methods in engineering analysis. The 52 participants included mathematicians and engineers with shared interest on discontinuous Galerkin or Petrov-Galerkin methods and other generalized nonconforming or mixed finite element methods

Adaptive modeling of plate structures

V disertaciji se ukvarjamo z različnimi vidiki modeliranja ploskovnih konstrukcij s končnimi elementi. Modeliranje plošč je nekoliko specifično in je zaradi kompleksnosti in pojavov, ki jih opisuje, bistveno prispevalo k razvoju same metode končnih elementov. Danes je na voljo več uveljavljenih modelov plošč in pripadajočih končnih elementov, ki uporabniku nudijo široko množico možnosti, iz katere lahko izbira. Prav široka možnost izbire predstavlja tudi največjo težavo, saj je težje določiti, kateri model je primernejši in tudi, katera mreža končnih elementov je za dan problem optimalna. Glavni cilj disertacije je raziskati ključne korake v procesu prilagodljivega modeliranja plošč, ki omogoča samodejno določitev optimalnega modela za dan problem. Ker je prilagodljivo modeliranje odvisno od zanesljivih ocen napak, je večji del disertacije posvečen metodam za izračun diskretizacijske in modelske napake. Na praktičnih primerih smo preučili nekaj najbolj uveljavljenih metod za oceno napake. V nasprotju z ocenami napake diskretizacije, je modelsko napako mnogo težje določiti. Posebna pozornost je bila zato namenjena metodi uravnoteženja rezidualov, ki ima potencial tudi na področju ocene modelske napake. V tem smislu to delo predstavlja pomemben prispevek k področju računanja modelske napake za plošče. Koncept prilagodljivega modeliranja ploskovnih konstrukcij je bil preskušen na hierarhični družini končnih elementov za plošče - od tankih plošč do modelov višjega reda, ki upoštevajo deformacije po debelini. Ravno dobro vzpostavljena hierarhija v družini končnih elementov se je pokazala za ključno pri zanesljivi oceni modelske napake. Prilagodljivo modeliranje ploskovnih konstrukcije je bilo preskušeno na nekaj zahtevnejših primerih. Območje je bilo najprej modeliranjo z najbolj grobim modelom na sorazmerno redki mreži. Z uporabo informacije o napaki začetnega izračuna je bil zgrajen nov model. Primerjava izračuna na novem modelu z začetnim računom je pokazala, da je predlagan način prilagodljivega modeliranja sposoben nadzorovati porazdelitev napake, kakor tudi zajeti vse pomembnejše po- jave, ki so značilni za modeliranje plošč.The thesis deals with adaptive finite element modeling of plate structures. The finite element modeling of plates has grown to a mature research topic, which has contributed significantly to the development of the finite element method for structural analysis due to its complexity and inherently specific issues. At present, several validated plate models and corresponding families of working and efficient finite elements are available, offering a sound basis for an engineer to choose from. In our opinion, the main problems in the finite modeling of plates are nowadays related to the adaptive modeling. Adaptive modeling should reach much beyond standard discretization (finite element mesh) error estimates and related mesh (discretization) adaptivity. It should also include model error estimates and model adaptivity, which should provide the most appropriate mathematical model for a specific region of a structure. Thus in this work we study adaptive modeling for the case of plates. The primary goal of the thesis is to provide some answers to the questions related to the key steps in the process of adaptive modeling of plates. Since the adaptivity depends on reliable error estimates, a large part of the thesis is related to the derivation of computational procedures for discretization error estimates as well as model error estimates. A practical comparison of some of the established discretization error estimates is made. Special attention is paid to what is called equilibrated residuum method, which has a potential to be used both for discretization error and model error estimates. It should be emphasized that the model error estimates are quite hard to obtain, in contrast to the discretization error estimates. The concept of model adaptivity for plates is in this work implemented on the basis of equilibrated residuum method and hierarchic family of plate finite element models. The finite elements used in the thesis range from thin plate finite elements to thick plate finite elements. The latter are based on a newly derived higher order plate theory, which includes through the thickness stretching. The model error is estimated by local element-wise compu- tations. As all the finite elements, representing the chosen plate mathematical models, are re-derived in order to share the same interpolation bases, the difference between the local com- putations can be attributed mainly to the model error. This choice of finite elements enables effective computation of the model error estimate and improves the robustness of the adaptive modeling. Thus the discretization error can be computed by an independent procedure. Many numerical examples are provided as an illustration of performance of the derived plate elements, the derived discretization error procedures and the derived modeling error procedure. Since the basic goal of modeling in engineering is to produce an effective model, which will produce the most accurate results with the minimum input data, the need for the adaptive modeling will always be present. In this view, the present work is a contribution to the final goal of the finite element modeling of plate structures: a fully automatic adaptive procedure for the construction of an optimal computational model (an optimal finite element mesh and an optimal choice of a plate model for each element of the mesh) for a given plate structure. vii

Applications of Asymptotic Analysis

This workshop focused on asymptotic analysis and its fundamental role in the derivation and understanding of the nonlinear structure of mathematical models in various fields of applications, its impact on the development of new numerical methods and on other fields of applied mathematics such as shape optimization. This was complemented by a review as well as the presentation of some of the latest developments of singular perturbation methods

Efficient finite element methods for aircraft engine noise prediction

Aircraft noise has a negative environmental impact. One of the ways in which it can be mitigated is by placing acoustic liners inside the aircraft's engines. These liners can be optimised for noise reduction. A cost effective way to optimise acoustic liners is to make use of numerical modelling. However, there is room for improvement of the efficiency of current modelling methods. This thesis is concerned with the efficient numerical prediction of noise emitted from modern aircraft engines. Four high order finite element methods are used to solve the convected wave equation, and their performances are compared. The benefit of using the hierarchic Lobatto finite element method to solve this type of problem is demonstrated. A scheme which optimises the efficiency of the high order method is developed. The scheme automatically chooses the most efficient order for a given element, depending on the element size, and the problem parameters on that element. The computational cost of using the standard quadratic finite element method to solve a typical engine intake noise problem, is compared to the cost of the proposed adaptive-order method. A significant improvement in terms of efficiency is demonstrated when using the proposed method over the standard method. Furthermore, a new formulation based on potential flow theory for the solution of vortex sheet problems (typically encountered when dealing with exhaust noise problems) is presented.

Workshop on the Integration of Finite Element Modeling with Geometric Modeling

The workshop on the Integration of Finite Element Modeling with Geometric Modeling was held on 12 May 1987. It was held to discuss the geometric modeling requirements of the finite element modeling process and to better understand the technical aspects of the integration of these two areas. The 11 papers are presented except for one for which only the abstract is given

A posteriori analysis for dynamic model adaptation in convection dominated problems

In this work we present an a posteriori error indicator for approximation schemes of Runge--Kutta--discontinuous--Galerkin type arising in applications of compressible fluid flows. The purpose of this indicator is not only for mesh adaptivity, we also make use of this to drive model adaptivity. This is where a perhaps costly complex model and a cheaper simple model are solved over different parts of the domain. The a posteriori bound we derive indicates the regions where the complex model can be relatively well approximated with the cheaper one. One such example which we choose to highlight is that of the Navier--Stokes--Fourier equations approximated by Euler's equations

A posteriori analysis for dynamic model adaptation in convection dominated problems

In this work we present an a posteriori error indicator for approximation schemes of Runge-Kutta-discontinuous-Galerkin type arising in applications of compressible fluid flows. The purpose of this indicator is not only for mesh adaptivity, we also make use of this to drive model adaptivity. This is where a perhaps costly complex model and a cheaper simple model are solved over different parts of the domain. The a posteriori bound we derive indicates the regions where the complex model can be relatively well approximated with the cheaper one. One such example which we choose to highlight is that of the Navier-Stokes-Fourier equations approximated by Euler's equations

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

Computational methods in electromagnetic biomedical inverse problems

This work concerns computational methods in electromagnetic biomedical inverse problems. The following biomedical imaging modalities are studied: electro/magnetoencephalography (EEG/MEG), electrical impedance tomography (EIT), and limited-angle computerized tomography (limited-angle CT). The use of a priori information about the unknown feature is necessary for finding an adequate answer to an inverse problem. Both classical regularization techniques and Bayesian methodology are applied to utilize the a priori knowledge. The inverse problems specifically considered in this work include determination of relatively small electric conductivity anomalies in EIT, dipole-like sources in EEG/MEG, and multiscale X-ray absorbing structures in limited-angle CT. Computational methods and techniques applied for solving these have been designed case-by-case. Results concern, among others, appropriate parametrization of inverse problems; two-stage reconstruction processes, in which a region of interest (ROI) is determined in the first stage and the actual reconstruction is found in the second stage; effective forward simulation through h- and p- versions of the finite element method (FEM); localization of dipole-like electric sources through hierarchical Bayesian models; implementation of the Kirsch factorization method for reconstruction of conductivity anomalies; as well as the use of a coarse-to-fine reconstruction strategy in linear inverse problems

Computational methods in cardiovascular mechanics

The introduction of computational models in cardiovascular sciences has been progressively bringing new and unique tools for the investigation of the physiopathology. Together with the dramatic improvement of imaging and measuring devices on one side, and of computational architectures on the other one, mathematical and numerical models have provided a new, clearly noninvasive, approach for understanding not only basic mechanisms but also patient-specific conditions, and for supporting the design and the development of new therapeutic options. The terminology in silico is, nowadays, commonly accepted for indicating this new source of knowledge added to traditional in vitro and in vivo investigations. The advantages of in silico methodologies are basically the low cost in terms of infrastructures and facilities, the reduced invasiveness and, in general, the intrinsic predictive capabilities based on the use of mathematical models. The disadvantages are generally identified in the distance between the real cases and their virtual counterpart required by the conceptual modeling that can be detrimental for the reliability of numerical simulations.Comment: 54 pages, Book Chapte