Lecture 13: 2D Problems using CST
|
|
- Meghan Shepherd
- 5 years ago
- Views:
Transcription
1 Lecture D Problems using CST APL705 Finite Element Method Two-dimensional Problems using Constant Strain Triangles To formulate D problems we will follow similar steps as in the case of D FE modeling. Here the displacements, traclon, distributed body force are funclons of the posilon given by (x,y) The displacement vector is given by u=[u v Where u and v are x and y components of u Stresses and strains are σ = [σ x,σ y,τ xy The body force and traclon vectors are f = [ f x, f y ε = [ε x,ε y,γ xy T = [T x,t y
2 FE Modeling of D Problem For elemental volume calculalon we have V = tda Where t is the thickness normal to element (along z direclon) Units of bod force force per unit volume and that of traclon is force per unit area. The strain displacement relalon is ε = u x, v y, u y + v x The stress strain relalon is matrix σ = Dε T where D is the material D FE Modeling Two dimensional problems discussed here we have two degrees of freedom at each node as follows The global displacement vector is Q = [ Q T, Q, Q... Q N ] where N is the total number of dof The given D domain is triangulated first and the triangulalon data is stored as nodal coordinates and conneclvity informalon. The coordinates are stored in an NNx matrix where NN is total number of nodes in the triangulalon. Q i i y x Q i-
3 Triangulated D domains Here we show two examples triangular meshes. First one straight edges which is tria -ngulated evenly. The other domain with curved boundary D Element ConnecLvity Considering an element, the conneclvity of a typical element in the triangular mesh is shown here q 6 The element displacement vector is shown here as q = [q, q, q, q 4, q 5, q 6 A typical conneclvity table is given as Element no./ local nodes N (x,y) u q q q 4 The element displacement vector q can be extracted from global Q using conneclvity table v q 5 q
4 The Local and Global Connect The nodal coordinates shown here as (x,y ),(x,y ) and (x,y ) have a global correspondence through the conneclvity table The local representalon of nodal coordinates and degrees of freedom is a way of clearly represenlng the element characterislcs. k = L T k 'L Constant Strain Triangle Recall the shape funclons used to interpolate the nodal displacements in D problems. Here in D problems we determine the displacements inside an element from the nodal displacements using linear shape funclons. By inspeclon we see that N +N +N =. This means that they are not linearly independent. 4
5 D Shape funclons The plot of third funclon is given here Note that the funclon N is at node and 0 at nodes and. Similarly N and N are also have their values as at nodes and respeclvely and 0 at other nodes. RepresenLng the independent shape funclons by ξ and η, we have N = ξ N = η and N = ξ η Here ξ and η are called the natural coordinates. D Shape funclons Now we look at the analogy of D and D shape funclons In D case, x-coordinate mapped onto ξ coordinate and shape funclon is a funclon of ξ. Here in D problems, x, y coordinates map onto ξ, η and the shape funclon is a funclon of ξ and η. The shape funclons are now represented by area coordinates. A point P(x,y) divides the triangle into three areas A, A and A as shown. Now express the shape funclons as area ralos. N + N + N = A A + A A + A A = For every point inside element N +N +N = 5
6 Isoparametric RepresentaLon Now displacements inside the element are expressed in terms of the nodal values of the unknown displacement field. u = N (ξ,η)q + N (ξ,η)q + N (ξ,η)q 5 v = N (ξ,η)q + N (ξ,η)q 4 + N (ξ,η)q 6 Now using the definilons of shape funclons N = ξ N = η and N = ξ η We write the displacements u,v as u = (q q 5 )ξ + (q q 5 )η + q 5 v = (q q 6 )ξ + (q 4 q 6 )η + q 6 As in D we represent N as a matrix now. Isoparametric RepresentaLon As in D we represent N as a matrix now. N 0 N 0 N 0 N = 0 N 0 N 0 N The displacements in matrix form { u} = N We know that for an isoparametric representalon the coordinates inside the triangular element also can be represented in terms of the nodal coordinates as x = N (ξ,η)x + N (ξ,η)x + N (ξ,η)x y = N (ξ,η)y + N (ξ,η)y + N (ξ,η)y [ ]{ q} OR x = (x x )ξ + (x x )η + x y = (y y )ξ + (y y )η + y 6
7 Isoparametric RepresentaLon The (x,y) coordinates inside the elements are wrihen as x = (x x )ξ + (x x )η + x y = (y y )ξ + (y y )η + y x ij = x i x j) By changing the notalon as x = x ξ + x η + x y = y ξ + y η + y ( ) 7
Bilinear Quadrilateral (Q4): CQUAD4 in GENESIS
Bilinear Quadrilateral (Q4): CQUAD4 in GENESIS The Q4 element has four nodes and eight nodal dof. The shape can be any quadrilateral; we ll concentrate on a rectangle now. The displacement field in terms
More informationME FINITE ELEMENT ANALYSIS FORMULAS
ME 2353 - FINITE ELEMENT ANALYSIS FORMULAS UNIT I FINITE ELEMENT FORMULATION OF BOUNDARY VALUE PROBLEMS 01. Global Equation for Force Vector, {F} = [K] {u} {F} = Global Force Vector [K] = Global Stiffness
More informationCode No: RT41033 R13 Set No. 1 IV B.Tech I Semester Regular Examinations, November - 2016 FINITE ELEMENT METHODS (Common to Mechanical Engineering, Aeronautical Engineering and Automobile Engineering)
More informationMMJ1153 COMPUTATIONAL METHOD IN SOLID MECHANICS PRELIMINARIES TO FEM
B Course Content: A INTRODUCTION AND OVERVIEW Numerical method and Computer-Aided Engineering; Phsical problems; Mathematical models; Finite element method;. B Elements and nodes, natural coordinates,
More informationLecture 21: Isoparametric Formulation of Plane Elements.
6.6. Rectangular Plane Stress/Strain Element. he CS element uses a triangular shape. he 3 nodes of the CS element allow us to employ linear multivariate approximations to the u and v displacements. he
More informationEshan V. Dave, Secretary of M&FGM2006 (Hawaii) Research Assistant and Ph.D. Candidate. Glaucio H. Paulino, Chairman of M&FGM2006 (Hawaii)
Asphalt Pavement Aging and Temperature Dependent Properties through a Functionally Graded Viscoelastic Model I: Development, Implementation and Verification Eshan V. Dave, Secretary of M&FGM2006 (Hawaii)
More informationFinite Element Method in Geotechnical Engineering
Finite Element Method in Geotechnical Engineering Short Course on + Dynamics Boulder, Colorado January 5-8, 2004 Stein Sture Professor of Civil Engineering University of Colorado at Boulder Contents Steps
More informationFinite Element Method-Part II Isoparametric FE Formulation and some numerical examples Lecture 29 Smart and Micro Systems
Finite Element Method-Part II Isoparametric FE Formulation and some numerical examples Lecture 29 Smart and Micro Systems Introduction Till now we dealt only with finite elements having straight edges.
More informationTriangular Plate Displacement Elements
Triangular Plate Displacement Elements Chapter : TRIANGULAR PLATE DISPLACEMENT ELEMENTS TABLE OF CONTENTS Page. Introduction...................... Triangular Element Properties................ Triangle
More informationShape Function Generation and Requirements
Shape Function Generation and Requirements Requirements Requirements (A) Interpolation condition. Takes a unit value at node i, and is zero at all other nodes. Requirements (B) Local support condition.
More informationChapter 5 Structural Elements: The truss & beam elements
Institute of Structural Engineering Page 1 Chapter 5 Structural Elements: The truss & beam elements Institute of Structural Engineering Page 2 Chapter Goals Learn how to formulate the Finite Element Equations
More informationAnalytical Mechanics: Elastic Deformation
Analytical Mechanics: Elastic Deformation Shinichi Hirai Dept. Robotics, Ritsumeikan Univ. Shinichi Hirai (Dept. Robotics, Ritsumeikan Univ.) Analytical Mechanics: Elastic Deformation 1 / 60 Agenda Agenda
More informationBHAR AT HID AS AN ENGIN E ERI N G C O L L E G E NATTR A MPA LL I
BHAR AT HID AS AN ENGIN E ERI N G C O L L E G E NATTR A MPA LL I 635 8 54. Third Year M E C H A NICAL VI S E M ES TER QUE S T I ON B ANK Subject: ME 6 603 FIN I T E E LE ME N T A N A L YSIS UNI T - I INTRODUCTION
More informationFinite Element Modeling and Analysis. CE 595: Course Part 2 Amit H. Varma
Finite Element Modeling and Analysis CE 595: Course Part 2 Amit H. Varma Discussion of planar elements Constant Strain Triangle (CST) - easiest and simplest finite element Displacement field in terms of
More informationInstitute of Structural Engineering Page 1. Method of Finite Elements I. Chapter 2. The Direct Stiffness Method. Method of Finite Elements I
Institute of Structural Engineering Page 1 Chapter 2 The Direct Stiffness Method Institute of Structural Engineering Page 2 Direct Stiffness Method (DSM) Computational method for structural analysis Matrix
More informationHIGHER-ORDER THEORIES
HIGHER-ORDER THEORIES Third-order Shear Deformation Plate Theory Displacement and strain fields Equations of motion Navier s solution for bending Layerwise Laminate Theory Interlaminar stress and strain
More informationChapter 6 2D Elements Plate Elements
Institute of Structural Engineering Page 1 Chapter 6 2D Elements Plate Elements Method of Finite Elements I Institute of Structural Engineering Page 2 Continuum Elements Plane Stress Plane Strain Toda
More informationStress analysis of a stepped bar
Stress analysis of a stepped bar Problem Find the stresses induced in the axially loaded stepped bar shown in Figure. The bar has cross-sectional areas of A ) and A ) over the lengths l ) and l ), respectively.
More informationLecture 27 Introduction to finite elements methods
Fall, 2017 ME 323 Mechanics of Materials Lecture 27 Introduction to finite elements methods Reading assignment: News: Instructor: Prof. Marcial Gonzalez Last modified: 10/24/17 7:02:00 PM Finite element
More informationMIXED FINITE ELEMENTS FOR PLATES. Ricardo G. Durán Universidad de Buenos Aires
MIXED FINITE ELEMENTS FOR PLATES Ricardo G. Durán Universidad de Buenos Aires - Necessity of 2D models. - Reissner-Mindlin Equations. - Finite Element Approximations. - Locking. - Mixed interpolation or
More informationApplications in Solid Mechanics
Companies, 4 CHAPTER 9 Applications in Solid 9. INTRODUCTION The bar and beam elements discussed in Chapters 4 are line elements, as only a single coordinate axis is required to define the element reference
More information3. Numerical integration
3. Numerical integration... 3. One-dimensional quadratures... 3. Two- and three-dimensional quadratures... 3.3 Exact Integrals for Straight Sided Triangles... 5 3.4 Reduced and Selected Integration...
More informationBACKGROUNDS. Two Models of Deformable Body. Distinct Element Method (DEM)
BACKGROUNDS Two Models of Deformable Body continuum rigid-body spring deformation expressed in terms of field variables assembly of rigid-bodies connected by spring Distinct Element Method (DEM) simple
More informationHIGHER-ORDER THEORIES
HIGHER-ORDER THEORIES THIRD-ORDER SHEAR DEFORMATION PLATE THEORY LAYERWISE LAMINATE THEORY J.N. Reddy 1 Third-Order Shear Deformation Plate Theory Assumed Displacement Field µ u(x y z t) u 0 (x y t) +
More informationMAE 323: Chapter 6. Structural Models
Common element types for structural analyis: oplane stress/strain, Axisymmetric obeam, truss,spring oplate/shell elements o3d solid ospecial: Usually used for contact or other constraints What you need
More informationGEO E1050 Finite Element Method Autumn Lecture. 9. Nonlinear Finite Element Method & Summary
GEO E1050 Finite Element Method Autumn 2016 Lecture. 9. Nonlinear Finite Element Method & Summary To learn today The lecture should give you overview of how non-linear problems in Finite Element Method
More informationMinimal Element Interpolation in Functions of High-Dimension
Minimal Element Interpolation in Functions of High-Dimension J. D. Jakeman, A. Narayan, D. Xiu Department of Mathematics Purdue University West Lafayette, Indiana Random Model as a function We can consider
More informationNumerical methods for PDEs FEM convergence, error estimates, piecewise polynomials
Platzhalter für Bild, Bild auf Titelfolie hinter das Logo einsetzen Numerical methods for PDEs FEM convergence, error estimates, piecewise polynomials Dr. Noemi Friedman Contents of the course Fundamentals
More informationPerformance comparison between hybridizable DG and classical DG methods for elastic waves simulation in harmonic domain
March 4-5, 2015 Performance comparison between hybridizable DG and classical DG methods for elastic waves simulation in harmonic domain M. Bonnasse-Gahot 1,2, H. Calandra 3, J. Diaz 1 and S. Lanteri 2
More informationJEPPIAAR ENGINEERING COLLEGE
JEPPIAAR ENGINEERING COLLEGE Jeppiaar Nagar, Rajiv Gandhi Salai 600 119 DEPARTMENT OFMECHANICAL ENGINEERING QUESTION BANK VI SEMESTER ME6603 FINITE ELEMENT ANALYSIS Regulation 013 SUBJECT YEAR /SEM: III
More informationSteps in the Finite Element Method. Chung Hua University Department of Mechanical Engineering Dr. Ching I Chen
Steps in the Finite Element Method Chung Hua University Department of Mechanical Engineering Dr. Ching I Chen General Idea Engineers are interested in evaluating effects such as deformations, stresses,
More informationIV B.Tech. I Semester Supplementary Examinations, February/March FINITE ELEMENT METHODS (Mechanical Engineering) Time: 3 Hours Max Marks: 80
www..com www..com Code No: M0322/R07 Set No. 1 IV B.Tech. I Semester Supplementary Examinations, February/March - 2011 FINITE ELEMENT METHODS (Mechanical Engineering) Time: 3 Hours Max Marks: 80 Answer
More informationNONLINEAR CONTINUUM FORMULATIONS CONTENTS
NONLINEAR CONTINUUM FORMULATIONS CONTENTS Introduction to nonlinear continuum mechanics Descriptions of motion Measures of stresses and strains Updated and Total Lagrangian formulations Continuum shell
More informationLecture 2: Finite Elements
Materials Science & Metallurgy Master of Philosophy, Materials Modelling, Course MP7, Finite Element Analysis, H. K. D. H. Bhadeshia Lecture 2: Finite Elements In finite element analysis, functions of
More informationInstitute of Structural Engineering Page 1. Method of Finite Elements I. Chapter 2. The Direct Stiffness Method. Method of Finite Elements I
Institute of Structural Engineering Page 1 Chapter 2 The Direct Stiffness Method Institute of Structural Engineering Page 2 Direct Stiffness Method (DSM) Computational method for structural analysis Matrix
More informationThe Plane Stress Problem
The Plane Stress Problem Martin Kronbichler Applied Scientific Computing (Tillämpad beräkningsvetenskap) February 2, 2010 Martin Kronbichler (TDB) The Plane Stress Problem February 2, 2010 1 / 24 Outline
More information. D CR Nomenclature D 1
. D CR Nomenclature D 1 Appendix D: CR NOMENCLATURE D 2 The notation used by different investigators working in CR formulations has not coalesced, since the topic is in flux. This Appendix identifies the
More informationCIVL4332 L1 Introduction to Finite Element Method
CIVL L Introduction to Finite Element Method CIVL L Introduction to Finite Element Method by Joe Gattas, Faris Albermani Introduction The FEM is a numerical technique for solving physical problems such
More informationA Basic Primer on the Finite Element Method
A Basic Primer on the Finite Element Method C. Berdin A. Rossoll March 1st 2002 1 Purpose Complex geometry and/or boundary conditions Local solution Non-linearities: geometric (large deformations/displacements)
More informationCubic Splines; Bézier Curves
Cubic Splines; Bézier Curves 1 Cubic Splines piecewise approximation with cubic polynomials conditions on the coefficients of the splines 2 Bézier Curves computer-aided design and manufacturing MCS 471
More informationME 1401 FINITE ELEMENT ANALYSIS UNIT I PART -A. 2. Why polynomial type of interpolation functions is mostly used in FEM?
SHRI ANGALAMMAN COLLEGE OF ENGINEERING AND TECHNOLOGY (An ISO 9001:2008 Certified Institution) SIRUGANOOR, TIRUCHIRAPPALLI 621 105 Department of Mechanical Engineering ME 1401 FINITE ELEMENT ANALYSIS 1.
More informationEFFECTS OF THERMAL STRESSES AND BOUNDARY CONDITIONS ON THE RESPONSE OF A RECTANGULAR ELASTIC BODY MADE OF FGM
Proceedings of the International Conference on Mechanical Engineering 2007 (ICME2007) 29-31 December 2007, Dhaka, Bangladesh ICME2007-AM-76 EFFECTS OF THERMAL STRESSES AND BOUNDARY CONDITIONS ON THE RESPONSE
More informationThe following syntax is used to describe a typical irreducible continuum element:
ELEMENT IRREDUCIBLE T7P0 command.. Synopsis The ELEMENT IRREDUCIBLE T7P0 command is used to describe all irreducible 7-node enhanced quadratic triangular continuum elements that are to be used in mechanical
More informationPrepared by M. GUNASHANKAR AP/MECH DEPARTMENT OF MECHANICAL ENGINEERING
CHETTINAD COLLEGE OF ENGINEERING AND TECHNOLOGY-KARUR FINITE ELEMENT ANALYSIS 2 MARKS QUESTIONS WITH ANSWER Prepared by M. GUNASHANKAR AP/MECH DEPARTMENT OF MECHANICAL ENGINEERING FINITE ELEMENT ANALYSIS
More informationScientific Computing WS 2018/2019. Lecture 15. Jürgen Fuhrmann Lecture 15 Slide 1
Scientific Computing WS 2018/2019 Lecture 15 Jürgen Fuhrmann juergen.fuhrmann@wias-berlin.de Lecture 15 Slide 1 Lecture 15 Slide 2 Problems with strong formulation Writing the PDE with divergence and gradient
More informationPost Graduate Diploma in Mechanical Engineering Computational mechanics using finite element method
9210-220 Post Graduate Diploma in Mechanical Engineering Computational mechanics using finite element method You should have the following for this examination one answer book scientific calculator No
More informationInterpolation (Shape Functions)
Mètodes Numèrics: A First Course on Finite Elements Interpolation (Shape Functions) Following: Curs d Elements Finits amb Aplicacions (J. Masdemont) http://hdl.handle.net/2099.3/36166 Dept. Matemàtiques
More informationABHELSINKI UNIVERSITY OF TECHNOLOGY
ABHELSINKI UNIVERSITY OF TECHNOLOGY TECHNISCHE UNIVERSITÄT HELSINKI UNIVERSITE DE TECHNOLOGIE D HELSINKI A posteriori error analysis for the Morley plate element Jarkko Niiranen Department of Structural
More informationModule I: Two-dimensional linear elasticity. application notes and tutorial. Problems
Module I: Two-dimensional linear elasticity application notes and tutorial Problems 53 selected excerpts from Read Me file for: ElemFin 1.1.1 Yannick CALLAUD in Symantec C++. 1 place of Falleron, 44300
More informationAn extended Galerkin weak form and a point interpolation method with continuous strain field and superconvergence using triangular mesh
Comput Mech (2009) 43:651 673 DOI 10.1007/s00466-008-0336-5 ORIGINAL PAPER An extended Galerkin weak form and a point interpolation method with continuous strain field and superconvergence using triangular
More informationLecture 12: Finite Elements
Materials Science & Metallurgy Part III Course M6 Computation of Phase Diagrams H. K. D. H. Bhadeshia Lecture 2: Finite Elements In finite element analysis, functions of continuous quantities such as temperature
More informationModule 2: Thermal Stresses in a 1D Beam Fixed at Both Ends
Module 2: Thermal Stresses in a 1D Beam Fixed at Both Ends Table of Contents Problem Description 2 Theory 2 Preprocessor 3 Scalar Parameters 3 Real Constants and Material Properties 4 Geometry 6 Meshing
More informationIntrinsic finite element modeling of a linear membrane shell problem
arxiv:3.39v [math.na] 5 Mar Intrinsic finite element modeling of a linear membrane shell problem Peter Hansbo Mats G. Larson Abstract A Galerkin finite element method for the membrane elasticity problem
More informationChapter 5. Introduction. Introduction. Introduction. Finite Element Modelling. Finite Element Modelling
Chaptr 5 wo-dimnsional problms using Constant Strain riangls (CS) Lctur Nots Dr Mohd Andi Univrsiti Malasia Prlis EN7 Finit Elmnt Analsis Introction wo-dimnsional init lmnt ormulation ollows th stps usd
More informationUNIVERSITY OF HAWAII COLLEGE OF ENGINEERING DEPARTMENT OF CIVIL AND ENVIRONMENTAL ENGINEERING
UNIVERSITY OF HAWAII COLLEGE OF ENGINEERING DEPARTMENT OF CIVIL AND ENVIRONMENTAL ENGINEERING ACKNOWLEDGMENTS This report consists of the dissertation by Ms. Yan Jane Liu, submitted in partial fulfillment
More informationENGN 2340 Final Project Report. Optimization of Mechanical Isotropy of Soft Network Material
ENGN 2340 Final Project Report Optimization of Mechanical Isotropy of Soft Network Material Enrui Zhang 12/15/2017 1. Introduction of the Problem This project deals with the stress-strain response of a
More informationStretching of a Prismatic Bar by its Own Weight
1 APES documentation (revision date: 12.03.10) Stretching of a Prismatic Bar by its Own Weight. This sample analysis is provided in order to illustrate the correct specification of the gravitational acceleration
More informationNumerical Solutions to Partial Differential Equations
Numerical Solutions to Partial Differential Equations Zhiping Li LMAM and School of Mathematical Sciences Peking University Variational Problems of the Dirichlet BVP of the Poisson Equation 1 For the homogeneous
More informationAnalytical Mechanics: Elastic Deformation
Analytical Mechanics: Elastic Deformation Shinichi Hirai Dept. Robotics, Ritsumeikan Univ. Shinichi Hirai (Dept. Robotics, Ritsumeikan Univ.) Analytical Mechanics: Elastic Deformation / 59 Agenda Agenda
More informationMEC-E8001 FINITE ELEMENT ANALYSIS
MEC-E800 FINIE EEMEN ANAYSIS 07 - WHY FINIE EEMENS AND IS HEORY? Design of machines and structures: Solution to stress or displacement by analytical method is often impossible due to complex geometry,
More informationSome improvements of Xfem for cracked domains
Some improvements of Xfem for cracked domains E. Chahine 1, P. Laborde 2, J. Pommier 1, Y. Renard 3 and M. Salaün 4 (1) INSA Toulouse, laboratoire MIP, CNRS UMR 5640, Complexe scientifique de Rangueil,
More informationSimulation in Computer Graphics Elastic Solids. Matthias Teschner
Simulation in Computer Graphics Elastic Solids Matthias Teschner Outline Introduction Elastic forces Miscellaneous Collision handling Visualization University of Freiburg Computer Science Department 2
More informationNUMERICAL ANALYSIS OF A PILE SUBJECTED TO LATERAL LOADS
IGC 009, Guntur, INDIA NUMERICAL ANALYSIS OF A PILE SUBJECTED TO LATERAL LOADS Mohammed Younus Ahmed Graduate Student, Earthquake Engineering Research Center, IIIT Hyderabad, Gachibowli, Hyderabad 3, India.
More informationNumerical Solutions to Partial Differential Equations
Numerical Solutions to Partial Differential Equations Zhiping Li LMAM and School of Mathematical Sciences Peking University The Residual and Error of Finite Element Solutions Mixed BVP of Poisson Equation
More informationPlane and axisymmetric models in Mentat & MARC. Tutorial with some Background
Plane and axisymmetric models in Mentat & MARC Tutorial with some Background Eindhoven University of Technology Department of Mechanical Engineering Piet J.G. Schreurs Lambèrt C.A. van Breemen March 6,
More informationA posteriori error estimation for elliptic problems
A posteriori error estimation for elliptic problems Praveen. C praveen@math.tifrbng.res.in Tata Institute of Fundamental Research Center for Applicable Mathematics Bangalore 560065 http://math.tifrbng.res.in
More informationDynamic Analysis of Laminated Composite Plate Structure with Square Cut-Out under Hygrothermal Load
Dynamic Analysis of Laminated Composite Plate Structure with Square Cut-Out under Hygrothermal Load Arun Mukherjee 1, Dr. Sreyashi Das (nee Pal) 2 and Dr. A. Guha Niyogi 3 1 PG student, 2 Asst. Professor,
More informationCHAPTER 8: Thermal Analysis
CHAPER 8: hermal Analysis hermal Analysis: calculation of temperatures in a solid body. Magnitude and direction of heat flow can also be calculated from temperature gradients in the body. Modes of heat
More informationInterpolation in h-version finite element spaces
Interpolation in h-version finite element spaces Thomas Apel Institut für Mathematik und Bauinformatik Fakultät für Bauingenieur- und Vermessungswesen Universität der Bundeswehr München Chemnitzer Seminar
More informationLecture 28 Introduction to finite elements methods
Fall, 2017 ME 323 Mechanics of Materials Lecture 28 Introduction to finite elements methods Reading assignment: News: Instructor: Prof. Marcial Gonzalez Last modified: 10/27/17 10:56:52 AM Some announcements
More informationMath 660-Lecture 15: Finite element spaces (I)
Math 660-Lecture 15: Finite element spaces (I) (Chapter 3, 4.2, 4.3) Before we introduce the concrete spaces, let s first of all introduce the following important lemma. Theorem 1. Let V h consists of
More informationDiscretization Methods Exercise # 5
Discretization Methods Exercise # 5 Static calculation of a planar truss structure: a a F Six steps: 1. Discretization 2. Element matrices 3. Transformation 4. Assembly 5. Boundary conditions 6. Solution
More informationNonconservative Loading: Overview
35 Nonconservative Loading: Overview 35 Chapter 35: NONCONSERVATIVE LOADING: OVERVIEW TABLE OF CONTENTS Page 35. Introduction..................... 35 3 35.2 Sources...................... 35 3 35.3 Three
More informationCIV-E1060 Engineering Computation and Simulation Examination, December 12, 2017 / Niiranen
CIV-E16 Engineering Computation and Simulation Examination, December 12, 217 / Niiranen This examination consists of 3 problems rated by the standard scale 1...6. Problem 1 Let us consider a long and tall
More informationTHE UPPER BOUND PROPERTY FOR SOLID MECHANICS OF THE LINEARLY CONFORMING RADIAL POINT INTERPOLATION METHOD (LC-RPIM)
International Journal of Computational Methods Vol. 4, No. 3 (2007) 521 541 c World Scientific Publishing Company THE UPPER BOUND PROPERTY FOR SOLID MECHANICS OF THE LINEARLY CONFORMING RADIAL POINT INTERPOLATION
More informationGyroscopic matrixes of the straight beams and the discs
Titre : Matrice gyroscopique des poutres droites et des di[...] Date : 29/05/2013 Page : 1/12 Gyroscopic matrixes of the straight beams and the discs Summarized: This document presents the formulation
More informationFinite Element Analysis Prof. Dr. B. N. Rao Department of Civil Engineering Indian Institute of Technology, Madras. Lecture - 06
Finite Element Analysis Prof. Dr. B. N. Rao Department of Civil Engineering Indian Institute of Technology, Madras Lecture - 06 In the last lecture, we have seen a boundary value problem, using the formal
More informationCIVL 8/7117 Chapter 12 - Structural Dynamics 1/75. To discuss the dynamics of a single-degree-of freedom springmass
CIV 8/77 Chapter - /75 Introduction To discuss the dynamics of a single-degree-of freedom springmass system. To derive the finite element equations for the time-dependent stress analysis of the one-dimensional
More informationInterpolation Functions for General Element Formulation
CHPTER 6 Interpolation Functions 6.1 INTRODUCTION The structural elements introduced in the previous chapters were formulated on the basis of known principles from elementary strength of materials theory.
More informationDevelopment of discontinuous Galerkin method for linear strain gradient elasticity
Development of discontinuous Galerkin method for linear strain gradient elasticity R Bala Chandran Computation for Design and Optimizaton Massachusetts Institute of Technology Cambridge, MA L. Noels* Aerospace
More informationRobust exponential convergence of hp-fem for singularly perturbed systems of reaction-diffusion equations
Robust exponential convergence of hp-fem for singularly perturbed systems of reaction-diffusion equations Christos Xenophontos Department of Mathematics and Statistics University of Cyprus joint work with
More informationInverse Design (and a lightweight introduction to the Finite Element Method) Stelian Coros
Inverse Design (and a lightweight introduction to the Finite Element Method) Stelian Coros Computational Design Forward design: direct manipulation of design parameters Level of abstraction Exploration
More informationLecture 15: Revisiting bars and beams.
3.10 Potential Energy Approach to Derive Bar Element Equations. We place assumptions on the stresses developed inside the bar. The spatial stress and strain matrices become very sparse. We add (ad-hoc)
More informationThe Finite Element Method
The Finite Element Method 3D Problems Heat Transfer and Elasticity Read: Chapter 14 CONTENTS Finite element models of 3-D Heat Transfer Finite element model of 3-D Elasticity Typical 3-D Finite Elements
More informationLinear Strain Triangle and other types of 2D elements. By S. Ziaei Rad
Linear Strain Triangle and other tpes o D elements B S. Ziaei Rad Linear Strain Triangle (LST or T6 This element is also called qadratic trianglar element. Qadratic Trianglar Element Linear Strain Triangle
More informationDHANALAKSHMI COLLEGE OF ENGINEERING, CHENNAI DEPARTMENT OF MECHANICAL ENGINEERING ME 6603 FINITE ELEMENT ANALYSIS PART A (2 MARKS)
DHANALAKSHMI COLLEGE OF ENGINEERING, CHENNAI DEPARTMENT OF MECHANICAL ENGINEERING ME 6603 FINITE ELEMENT ANALYSIS UNIT I : FINITE ELEMENT FORMULATION OF BOUNDARY VALUE PART A (2 MARKS) 1. Write the types
More informationA study of moderately thick quadrilateral plate elements based on the absolute nodal coordinate formulation
Multibody Syst Dyn (2014) 31:309 338 DOI 10.1007/s11044-013-9383-6 A study of moderately thick quadrilateral plate elements based on the absolute nodal coordinate formulation Marko K. Matikainen Antti
More informationTHE FINITE ELEMENT METHOD 2017 Dept. of Solid Mechanics
THE FINITE ELEMENT METHOD 07 Dept. of Solid Mechanics EXAMINATION: 07-05-9 A maximum of 60 points can be achieved in this examination. To pass at least 0 points are required. Permitted aid: Pocket calculator.
More informationMITOCW MITRES2_002S10nonlinear_lec15_300k-mp4
MITOCW MITRES2_002S10nonlinear_lec15_300k-mp4 The following content is provided under a Creative Commons license. Your support will help MIT OpenCourseWare continue to offer high quality educational resources
More informationContinuum Mechanics and the Finite Element Method
Continuum Mechanics and the Finite Element Method 1 Assignment 2 Due on March 2 nd @ midnight 2 Suppose you want to simulate this The familiar mass-spring system l 0 l y i X y i x Spring length before/after
More informationFinite Element Analysis of Saint-Venant Torsion Problem with Exact Integration of the Elastic-Plastic Constitutive
Finite Element Analysis of Saint-Venant Torsion Problem with Exact Integration of the Elastic-Plastic Constitutive Equations W. Wagner Institut für Baustatik Universität Karlsruhe (TH) Kaiserstraße 12
More informationInterpolation Models CHAPTER 3
CHAPTER 3 CHAPTER OUTLINE 3. Introduction 75 3.2 Polynomial Form of Interpolation Functions 77 3.3 Simple, Comple, and Multiple Elements 78 3.4 Interpolation Polynomial in Terms of Nodal Degrees of Freedom
More informationENGN2340 Final Project: Implementation of a Euler-Bernuolli Beam Element Michael Monn
ENGN234 Final Project: Implementation of a Euler-Bernuolli Beam Element Michael Monn 12/11/13 Problem Definition and Shape Functions Although there exist many analytical solutions to the Euler-Bernuolli
More informationPreprocessor Geometry Properties )Nodes, Elements(, Material Properties Boundary Conditions(displacements, Forces )
در برنامه يك تدوين براي بعدي دو يك سازه محيط MATLAB Preprocessor Geometry Properties )Nodes, Elements(, Material Properties Boundary Conditions(displacements, Forces ) Definition of Stiffness Matrices
More informationA first order divided difference
A first order divided difference For a given function f (x) and two distinct points x 0 and x 1, define f [x 0, x 1 ] = f (x 1) f (x 0 ) x 1 x 0 This is called the first order divided difference of f (x).
More informationFinite Element Analysis Prof. Dr. B. N. Rao Department of Civil Engineering Indian Institute of Technology, Madras. Module - 01 Lecture - 13
Finite Element Analysis Prof. Dr. B. N. Rao Department of Civil Engineering Indian Institute of Technology, Madras (Refer Slide Time: 00:25) Module - 01 Lecture - 13 In the last class, we have seen how
More informationNumerical Methods-Lecture VIII: Interpolation
Numerical Methods-Lecture VIII: Interpolation (See Judd Chapter 6) Trevor Gallen Fall, 2015 1 / 113 Motivation Most solutions are functions Many functions are (potentially) high-dimensional Want a way
More information2C9 Design for seismic and climate changes. Jiří Máca
2C9 Design for seismic and climate changes Jiří Máca List of lectures 1. Elements of seismology and seismicity I 2. Elements of seismology and seismicity II 3. Dynamic analysis of single-degree-of-freedom
More information1 Nonlinear deformation
NONLINEAR TRUSS 1 Nonlinear deformation When deformation and/or rotation of the truss are large, various strains and stresses can be defined and related by material laws. The material behavior can be expected
More informationCIVL 7/8117 Chapter 4 - Development of Beam Equations - Part 2 1/34. Chapter 4b Development of Beam Equations. Learning Objectives
CIV 7/87 Chapter 4 - Development of Beam Equations - Part /4 Chapter 4b Development of Beam Equations earning Objectives To introduce the work-equivalence method for replacing distributed loading by a
More information