Nonlinear finite elements

Visualization of how a car deforms in an asymmetrical crash using finite element analysis.

Welcome to this learning project about nonlinear finite elements!

Contents   [hide]  1 Introduction 1.1 Project metadata 1.2 Content summary 1.3 Goals 2 Contents 2.1 Syllabus and Learning Materials 2.2 Assignments 2.3 Tests and Quizzes 3 Textbooks and References 3.1 Textbooks 3.2 References 3.3 Reading List 3.4 External links

Introduction[edit]

Finite element analysis (FEA) is a computer simulation technique used in engineering analysis. It uses a numerical technique called thefinite element method (FEM). There are many finite element software packages, both free and proprietary. Development of the finite element method in structural mechanics is usually based on an energy principle such as the virtual work principle or the minimum total potential energy principle. *

Project metadata[edit]

Quality resource: this resource is afeatured learning resource.

Completion status: this resource has reached a high level ofcompletion.

Educational level: this is a tertiary(university) resource.

Subject classification: this is anengineering resource .

• Suggested Prerequisites:
  • Linear algebra
  • Partial differential equations
  • Introduction to finite elements
  • Continuum mechanics
• Time investment: 6 months
• Portal:Engineering and Technology
• School:Engineering
• Department:Mechanical engineering
• Level: First year graduate

Content summary[edit]

This is an introductory course on nonlinear finite element analysis of solid mechanics and heat transfer problems. Nonlinearities can be caused by changes in geometry or be due to nonlinear material behavior. Both types of nonlinearities are covered in this course.

Goals[edit]

This learning project aims to.

• provide the mathematical foundations of the finite element formulation for engineering applications (solids, heat, fluids).
• expose students to some of the recent trends and research areas in finite elements.

Here's a short quiz to help you find out what you need to brush up on before you dig into the course:

• Assessment Quiz

Contents[edit]

Syllabus and Learning Materials Syllabus and Learning Materials[edit] Mathematical Preliminaries Set notation Functions Vectors Matrices Tensors Partial differential equations Variational calculus Linear finite element basics An example: Axially loaded bar Strong form: governing differential equation Weak form: integral equation Approximate solution: Galerkin method Approximate solution: Finite element method More examples: Some model problems Weak form: Weighted residual methods Choosing a weight function Bubnov Galerkin methods Finite element basis functions Example of a finite element approximation A time-dependent problem: the heat equation Steady state heat conduction Weak form of the steady state heat equation Weak form of the time-dependent heat equation Finite element approximation of Poisson equation Finite element approximation of the heat equation Time integration of the heat equation. Nonlinear finite element basics Nonlinearities in solid mechanics An example: Nonlinear deformation of an axially loaded bar Weak form for the nonlinear bar The Newton-Raphson method The Newton method applied to finite elements The Newton method applied to the axially loaded bar Lagrangian and Eulerian descriptions of motion Lagrangian finite elements Motion from the Lagrangian point of view Total Lagrangian approach Updated Lagrangian approach An example: Effect of mesh distortion Solution procedure Special case: Natural vibrations Nonlinear deformation of beams Euler-Bernoulli beams Timoshenko beams Buckling of beams Nonlinear deformation of plates and shells Basic linear plate and shell elements Nonlinear plates and shells Time-dependent deformation of shells Basic continuum mechanics Kinematics Motion, displacement, velocity, acceleration Stresses and strains in one and two dimensions Strains and deformations in three-dimensions Polar decomposition Spectral decompositions of kinematic quantities Volume change and area change Time derivatives and rate quantities Objectivity of kinematic quantities Stress measures and stress rates Stress measures Deviatoric and volumetric stress Objective stress rates Balance laws Overview of balance laws Balance of mass Balance of linear momentum Balance of angular momentum Balance of energy Entropy inequality Constitutive models Objectivity of hyperelastic relations Rate form of hyperelastic relations Nonlinear elasticity Plasticity Viscoplasticity Viscoelasticity. Finite element formulation in three dimensions. Updated Lagrangian formulation Verification and Validation.

Assignments, tests and quizzes Assignments[edit] Homework 1 Problem set Solutions Homework 2 Problem set Solutions Homework 3 Problem set Solutions Homework 4 Problem set Hints Solutions Homework 5 Problem set Solutions Homework 6 Problem set Hints Solutions Homework 7 Problem set Hints Solutions Homework 8 Problem set Solutions Homework 9 Problem set Solutions Homework 10 Problem set Solutions Homework 11 Problem set Solutions Tests and Quizzes[edit] Quiz 1 Solutions

Textbooks and References[edit]

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Textbooks[edit]

• An Introduction to Nonlinear Finite Element Analysis by J. N. Reddy, Oxford University Press, 2004, ISBN 019852529X.
• Nonlinear Finite Elements for Continua and Structures by T. Belytschko, W. K. Liu, and B. Moran, John Wiley and Sons, 2000.
• Computational Inelasticity by J. C. Simo and T. J. R. Hughes, Springer, 1998.
• The Finite Element Method: Linear Static and Dynamic Finite Element Analysis by T. J. R. Hughes, Dover Publications, 2000.

References[edit]

• Finite Element Analysis:
  • K-J. Bathe (1996), Finite Element Procedures, Prentice-Hall. Useful repository of information on nonlinear finite elements.
  • J. N. Reddy (1993), An Introduction to the Finite Element Method, McGraw-Hill. This book is referred to a number of times in one of the texts.
  • O. C. Zienkiewicz and R. L. Taylor (2000), The Finite Element Method: Volume 2 Solid Mechanics, Butterworth-Heinemann. Another excellent repository of information of nonlinear finite elements geared toward the Civil Engineers.
  • S. C. Brenner and L. R. Scott (2007), The mathematical theory of finite element methods, vol. 15 of Texts in Applied Mathematics, Springer-Verlag, Classical book on the mathematics foundation of finite element methods.
• Continuum Mechanics:
  • R. M. Brannon (2004), Large Deformation Kinematics. An excellent introduction to kinematics.
  • R. M. Brannon (2004), Rotation. Almost everything you ever wanted to know about rotations.
  • A.J.M Spencer (2004), Continuum Mechanics, Dover Publications. An excellent introduction to continuum mechanics. You should own a copy for your personal library.
  • L.E. Malvern (1969), Introduction to the Mechanics of a Continuous Medium, Prentice-Hall. A good reference for continuum mechanics.
• Mathematics:
  • R. M. Brannon (2004), Elementary Vector and Tensor Analysis for Engineers. This free online book is the best introduction I have seen for vector and tensor analysis for nonlinear mechanics.
  • R. M. Brannon (2004), Curvilinear Coordinates. Very useful if you wish to follow the literature on the nonlinear deformation of shells.
  • B. Daya Reddy (1998), Introductory Functional Analysis: With applications to boundary value problems and finite elements. , Springer-Verlag. Excellent book for engineers who want to understand the terminology used in the finite element literature and how error analysis is done.
  • A.P.S. Selvadurai (2000), Partial Differential Equations in Mechanics 1,2. Springer. Excellent introductory text on partial differential equations with engineers in mind.

Reading List[edit]

• Taylor, R.L., Simo, J.C., Zienkiewicz, O.C., and Chan, A.C.H, 1986, The patch test - a condition for assessing FEM convergence,International Journal for Numerical Methods in Engineering, 22, pp. 39-62.
• Simo, J.C. and Vu-Quoc, L., 1986, A three-dimensional finite strain rod model. Part II: Computational Aspects, Computer Methods in Applied Mechanics and Engineering, 58, pp. 79-116.
• Ibrahimbegovic, A., 1995, On finite element implementation of geometrically nonlinear Reissner's beam theory: Three-dimensional curved beam elements, Computer Methods in Applied Mechanics and Engineering, 122, pp. 11-26.
• Buchter, N., Ramm, E., and Roehl, D., 1994, Three-dimensional extension of non-linear shell formulation based on the enhanced assumed strain concept, Int. J. Numer. Meth. Engng., 37, pp. 2551-2568.
• Rouainia, M. and Peric, D., 1998, A computational model for elasto-viscoplastic solids at finite strain with reference to thin shell applications, Int. J. Numer. Meth. Engng., 42, pp. 289-311.

External links[edit]

• Gmsh - GPL'd finite element analyzer

Hingga nonlinier elemen

Visualisasi tentang bagaimana mobil deformasi dalam kecelakaan asimetris menggunakan analisis elemen hingga.

Selamat Datang di proyek ini belajar tentang elemen hingga nonlinier!

Isi   [hide]  1 Pendahuluan 1,1 Proyek metadata 1,2 Konten ringkasan 1,3 Tujuan 2 Isi 2.1 Silabus dan Bahan Belajar 2.2 Tugas 2.3 Tes dan Kuis 3 Buku Teks dan Referensi 3.1 Buku teks 3.2 Referensi 3.3 Membaca Daftar 3.4 Pranala luar

Pendahuluan[ sunting ]

Analisis elemen hingga (FEA) adalah simulasi komputer teknik yang digunakan dalam analisis rekayasa. Ini menggunakan teknik numerik yang disebut metode elemen hingga (FEM). Ada banyak elemen hingga paket perangkat lunak , baik bebas dan kepemilikan .Pengembangan metode elemen hingga dalam mekanika struktur biasanya didasarkan pada prinsip energi seperti kerja virtual prinsip atauprinsip energi minimum total potensi . *

Proyek metadata[ sunting ]

Kualitas sumber daya : sumber daya ini adalah sumber belajar fitur .

Penyelesaian Status : sumber ini telah mencapai tingkat tinggi selesai .

Tingkat pendidikan : ini adalahtersier sumber daya (universitas).

Klasifikasi Subyek : ini adalah sebuah rekayasa sumber daya.

• Saran Prasyarat:
  • Linear aljabar
  • Partial persamaan diferensial
  • Pengantar elemen hingga
  • Continuum mekanika
• Waktu investasi: 6 bulan
• Portal : Rekayasa dan Teknologi
• Sekolah : Teknik
• Departemen: Teknik mesin
• Level: lulusan tahun Pertama

Ringkasan Konten[ sunting ]

Ini merupakan pengantar saja pada analisis elemen hingga nonlinier mekanika padat dan masalah perpindahan panas. Nonlinier dapat disebabkan oleh perubahan geometri atau disebabkan oleh perilaku material nonlinier. Kedua jenis nonlinier akan dibahas dalam kursus ini.

Tujuan[ sunting ]

Proyek ini bertujuan untuk belajar.

• memberikan dasar matematika formulasi elemen hingga untuk aplikasi teknik (solid, panas, cairan).
• mengekspos siswa untuk beberapa tren terbaru dan penelitian di bidang elemen hingga.

Berikut adalah kuis singkat untuk membantu Anda menemukan apa yang Anda butuhkan untuk menyegarkan kembali sebelum Anda menggali ke dalam kursus:

• Penilaian Kuis

Isi[ sunting ]

Silabus dan Materi Pembelajaran Silabus dan Materi Pembelajaran[ sunting ] Matematika Pendahuluan Set notasi Fungsi Vektor Matriks Tensor Partial persamaan diferensial Variational kalkulus Linear elemen hingga dasar-dasar Contoh: bar aksial dimuat Kuat bentuk: persamaan diferensial yang mengatur Lemah bentuk: persamaan integral Perkiraan solusi: metode Galerkin Perkiraan solusi: metode elemen hingga Lebih contoh: Beberapa masalah Model Lemah bentuk: metode sisa tertimbang Memilih fungsi bobot Bubnov Galerkin metode Dasar elemen fungsi terbatas Contoh pendekatan elemen hingga Masalah tergantung waktu: persamaan panas Steady state panas konduksi Lemahnya bentuk persamaan panas steady state Lemahnya bentuk persamaan panas tergantung waktu Pendekatan elemen hingga dari persamaan Poisson Pendekatan elemen hingga dari persamaan panas Waktu integrasi dari persamaan panas . Elemen hingga nonlinier dasar-dasar Nonlinier dalam mekanika padat Contoh: deformasi nonlinier dari bar dimuat secara aksial Lemah bentuk untuk bar nonlinier Metode Newton-Raphson Metode Newton diterapkan pada elemen hingga Metode Newton diterapkan ke bar secara aksial dimuat Lagrangian dan deskripsi Eulerian gerak Lagrangian terbatas elemen Gerakan dari sudut pandang Lagrangian Jumlah Lagrangian pendekatan Diperbarui Lagrangian pendekatan Contoh: Pengaruh mesh distorsi Solusi prosedur Khusus kasus: getaran Alam Nonlinier deformasi balok Euler-Bernoulli balok Timoshenko balok Tekuk balok Nonlinier deformasi pelat dan cangkang Dasar linear piring dan elemen shell Nonlinier pelat dan cangkang Tergantung waktu deformasi kerang Dasar mekanika kontinum Kinematika Gerak, perpindahan, kecepatan, percepatan Tegangan dan regangan dalam satu dan dua dimensi Strain dan deformasi dalam tiga-dimensi Polar dekomposisi Spektral dekomposisi dalam jumlah kinematik Volume perubahan dan perubahan daerah Waktu derivatif dan jumlah tingkat Objektivitas dalam jumlah kinematik Stres ukuran dan tingkat stres Stres tindakan Deviatorik dan volumetrik stres Tujuan tingkat stres Saldo hukum Tinjauan hukum keseimbangan Neraca massa Neraca momentum linier Neraca momentum sudut Neraca energi Entropi ketidaksetaraan Konstitutif model Objektivitas hubungan hyperelastic Tingkat bentuk hubungan hyperelastic Elastisitas nonlinier Keliatan Viscoplasticity Viscoelasticity. Elemen hingga formulasi dalam tiga dimensi. Diperbarui Lagrangian formulasi Verifikasi dan Validasi.