ABHELSINKI UNIVERSITY OF TECHNOLOGY
|
|
- Ashlynn Nelson
- 5 years ago
- Views:
Transcription
1 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 Engineering and Building Technology TKK Helsinki University of Technology, Finland Lourenço Beirão da Veiga, University of Milan, Italy Rolf Stenberg, Institute of Mathematics, TKK, Finland
2 Contents 1 Introduction 2 Kirchhoff plate bending model 3 Morley finite element formulation 4 A posteriori error estimates 5 Numerical results 6 Conclusions and Discussion References 2
3 Contents 1 Introduction 2 Kirchhoff plate bending model 3 Morley finite element formulation 4 A posteriori error estimates 5 Numerical results 6 Conclusions and Discussion References 3
4 Introduction Thin structures (shells, plates, membranes, beams) are the main building blocks in modern structural design. Beside the classical fields as civil engineering, thevarietyof applications have strongly increased also in many other fields as aeronautics, biomechanics, surgical medicine or microelectronics. In particular, new applications arise when thin structures are combined with functional, smart or composite materials (shape memory alloys, piezo-electric cheramics etc.). Increasing demands for accuracy and productivity have created a need for adaptive (automated, efficient, reliable) computational methods for thin structures. 4
5 Contents 1 Introduction 2 Kirchhoff plate bending model 3 Morley finite element formulation 4 A posteriori error estimates 5 Numerical results 6 Conclusions and Discussion References 5
6 Kirchhoff plate bending model We consider bending of a thin planar structure occupied by P =Ω ( t 2, t 2 ), where Ω R 2 denotes the midsurface of the plate and t diam(ω) denotes the thickness of the plate. Kinematical assumptions for the dimension reduction: Straight fibres normal to the midsurface remain straight and normal. Fibres normal to the midsurface do not stretch. The midsurface moves only in the vertical direction. 6
7 Deformations Under these assumptions, with the deflection w, the displacement field u =(u x,u y,u z ) takes the form u x = z w(x, y) x, u y = z w(x, y) y, u z = w(x, y). The corresponding deformation is defined by the symmetric linear strain tensor in the component form as e xx = z 2 w x 2, ε(u) = 1 2( u +( u) T ), e yy = z 2 w y 2, e zz =0, e xy = z 2 w x y, e xz =0, e yz =0. 7
8 Stress resultants Next, we define the stress resultants, the moments and the shear forces: M xy t/2 M = M xx with M ij = zσ ij dz, i, j = x, y, M yx M yy t/2 Q = Q t/2 x with Q i = σ iz dz, i = x, y, Q y t/2 where the stress tensor is assumed to be symmetric: σ ij = σ ji, i,j = x, y, z. 8
9 Equilibrium equations and boundary conditions The principle of virtual work gives, with the load resultant F,the equilibrium equation div div M = F with div M + Q = 0. and the boundary conditions w =0, w n =0 onγ C, w =0, n Mn =0 onγ S, n Mn =0, 2 s 2 (s Mn)+n div M =0 onγ F, (s 1 Mn 1 )(c) =(s 2 Mn 2 )(c) c V, where the indices 1 and 2 refer to the sides of the boundary angle at a corner point c on the free boundary Γ F. 9
10 Constitutive assumptions The material of the plate is assumed to be linearly elastic (defined by the generalized Hooke s law) homogeneous (independent of the coordinates x, y, z) isotropic (independent of the coordinate system). Furthermore, we assume that the transverse normal stress vanishes: σ zz =0. 10
11 Variational formulation Let the deflection w belong to the Sobolev space W = {v H 2 (Ω) v =0onΓ C Γ S, v n =0onΓ C }, where n indicates the unit outward normal to the boundary Γ. Problem. Variational formulation: Findw W such that (Eε( w), ε( v)) = (f,v) v W, with the elasticity tensor E defined as ( Eε = ε + E 12(1 + ν) ν ) 1 ν tr(ε)i ε R 2 2, with Young s modulus E and the Poisson ratio ν. 11
12 Morley finite element formulation Let E denote an edge of a triangle K in a triangulation T h. We define the discrete space for the deflection as W h = { v M 2,h v n E =0 E Eh i Eh} c, E where M 2,h denotes the space of the second order piecewise polynomial functions on T h which are continuous at the vertices of all the internal triangles and zero at all the triangle vertices of Γ C Γ S. Finite element method. Morley: Findw h W h such that K T h (Eε( w h ), ε( v)) K =(f,v) v W h. 12
13 A priori error estimate The method is stable and convergent with respect to the following discrete norm on the space W h + H 2 : v 2 h := K T h v 2 2,K + E E h h 3 E v 2 0,E + h 1 E E E h v n E 2 0,E, Proposition. (Shi 90, Ming and Xu 06) Assuming that Γ=Γ C there exists a positive constant C such that w w h h Ch ( w H3 (Ω) + h f L2 (Ω)). The numerical results indicate the same convergence rate for general boundary conditions as well. 13
14 Contents 1 Introduction 2 Kirchhoff plate bending model 3 Morley finite element formulation 4 A posteriori error estimates 5 Numerical results 6 Conclusions and Discussion References 14
15 A posteriori error estimates We use the following notation: for jumps (and traces), h E and h K for the edge length and the element diameter. Interior error indicators For all the elements K in the mesh T h, η 2 K := h 4 K f 2 0,K, and for all the internal edges E I h, η 2 E := h 3 E w h 2 0,E + h 1 E w h n E 2 0,E. 15
16 Boundary error indicators The boundary of the plate is divided into clamped, simply supported and free parts: Γ:= Ω =Γ C Γ S Γ F. For edges on the clamped and simply supported boundaries Γ C and boundary Γ S, respectively, ηe,c 2 := h 3 E w h 2 0,E + h 1 E w h 2 n 0,E, E ηe,s 2 := h 3 E w h 2 0,E. 16
17 Error indicators local and global For any element K T h,letthelocal error indicator be η K := ( η 2 K η 2 E + η 2 E,C + η 2 E,S) 1/2, E I h E K E C h E K E S h E K with the notation I h for the collection of all the internal edges, C h and S h for the collections of all the boundary edges on Γ C and Γ S, respectively. The global error indicator is defined as η h := ( K T h η 2 K ) 1/2. 17
18 Upper bound Reliability Theorem. Reliability: There exists a positive constant C such that w w h h Cη h. Lower bound Efficiency Theorem. Efficiency: For any element K, there exists a positive constant C K such that η K C K ( w wh h,k + h 2 K f f h 0,K ). Efficiency is proved by standard arguments; reliability needs a new Clément-type interpolant and a new Helmholtz-type decomposition. 18
19 Techniques for the analysis Helmholtz decomposition Lemma. Let σ be a second order tensor field in L 2 (Ω; R 2 2 ). Then, there exist ψ W, ρ L 2 0(Ω) and φ [ H 1 (Ω)] 2 such that σ = Eε( ψ)+ρ + Curl φ, with ρ = 0 ρ. ρ 0 ψ H2 (Ω) + ρ L2 (Ω) + φ H1 (Ω) C σ L2 (Ω). Here H m (Ω), m N, indicate the quotient space of H m (Ω) where the seminorm Hm (Ω) is null. In analysis, Lemma is applied to the tensor field Eε( (w w h )). 19
20 Contents 1 Introduction 2 Kirchhoff plate bending model 3 Morley finite element formulation 4 A posteriori error estimates 5 Numerical results 6 Conclusions and Discussion References 20
21 Numerical results We have implemented the method in the open-source finite element software Elmer developed by CSC the Finnish IT Center for Science. The software provides error balancing strategy and complete remeshing for triangular meshes. We have used test problems with convex rectangular domains and with known exact solutions for investigating the effectivity index for the error estimator derived. Non-convex domains we have used for studying the adaptive performance and robustness of the method. 21
22 Effectivity index ι = η h w w h h Effectivity Index = Error Estimator / Exact Error Number of Elements Effectivity Index = Error Estimator / Exact Error Number of Elements Figure 1: Left: uniform refinements; Right: adaptive refinements. Clamped (squares), simply supported (circles) and free (triangles) boundaries included. 22
23 Adaptively refined mesh Error estimator Simply supported L-corner Figure 2: Simply supported L-shaped domain: Distribution of the error estimator for two adaptive steps. 23
24 Uniform vs. Adaptive Convergence 10 1 Convergence of the Error Estimator Number of Elements Figure 3: Simply supported L-shaped domain: Convergence of the error estimator for the uniform refinements and adaptive refinements; Solid lines for global, dashed lines for maximum local ones. 24
25 Adaptively refined mesh Error estimator Clamped L-corner Figure 4: Simply supported L-shaped domain with a clamped L- corner: Distribution of the error estimator for two adaptive steps. 25
26 Uniform vs. Adaptive Convergence 10 1 Convergence of the Error Estimator Number of Elements Figure 5: Clamped L-corner: Convergence of the error estimator for the uniform refinements and adaptive refinements; Solid lines for global, dashed lines for maximum local ones. 26
27 Adaptively refined mesh Error estimator Simply supported M-domain Figure 6: Simply supported M-shaped domain: Distribution of the error estimator for two adaptive steps. 27
28 Uniform vs. Adaptive Convergence 10 1 Convergence of the Error Estimator Number of Elements Figure 7: Simply supported M-shaped domain: Convergence of the error estimator for the uniform refinements and adaptive refinements; Solid lines for global, dashed lines for maximum local ones. 28
29 Contents 1 Introduction 2 Kirchhoff plate bending model 3 Morley finite element formulation 4 A posteriori error estimates 5 Numerical results 6 Conclusions and Discussion References 29
30 Conclusions and Discussion Advantages Reliability: computable (non-guaranteed due to C) global upper bound for the error. Efficiency: computable (non-guaranteed due to C K )locallower bound. Robustness: C K independent of the mesh size, data and the solution. Computational costs: small (local) compared to solving the problem itself. 30
31 Disadvantages Residual based error estimates in the energy norm only no estimates for other quantities of interest. Method dependent: applicaple for the Morley element only although the techniques can be generalized. Valid only for static problem with transversal loading and isotropic, homogeneous, linearly elastic material sofar. 31
32 References [1] L. Beirão da Veiga, J. Niiranen, R. Stenberg: A posteriori error estimates for the Morley plate bending element; Numerische Matematik, 106, (2007). [2] L. Beirão da Veiga, J. Niiranen and R. Stenberg. A posteriori error analysis for the Morley plate element with general boundary conditions; Research Reports A556, Helsinki University of Technology, Institute of Mathematics, December submitted for publication. 32
ABHELSINKI UNIVERSITY OF TECHNOLOGY
ABHELSINKI UNIVERSITY OF TECHNOLOGY TECHNISCHE UNIVERSITÄT HELSINKI UNIVERSITE DE TECHNOLOGIE D HELSINKI A posteriori error analysis for Kirchhoff plate elements Jarkko Niiranen Laboratory of Structural
More information3D Elasticity Theory
3D lasticity Theory Many structural analysis problems are analysed using the theory of elasticity in which Hooke s law is used to enforce proportionality between stress and strain at any deformation level.
More informationENERGY NORM A POSTERIORI ERROR ESTIMATES FOR MIXED FINITE ELEMENT METHODS
ENERGY NORM A POSTERIORI ERROR ESTIMATES FOR MIXED FINITE ELEMENT METHODS CARLO LOVADINA AND ROLF STENBERG Abstract The paper deals with the a-posteriori error analysis of mixed finite element methods
More informationGustafsson, Tom; Stenberg, Rolf; Videman, Juha A posteriori analysis of classical plate elements
Powered by TCPDF (www.tcpdf.org) This is an electronic reprint of the original article. This reprint may differ from the original in pagination and typographic detail. Gustafsson, Tom; Stenberg, Rolf;
More informationA-priori and a-posteriori error estimates for a family of Reissner-Mindlin plate elements
A-priori and a-posteriori error estimates for a family of Reissner-Mindlin plate elements Joint work with Lourenco Beirão da Veiga (Milano), Claudia Chinosi (Alessandria) and Carlo Lovadina (Pavia) Previous
More informationBasic Equations of Elasticity
A Basic Equations of Elasticity A.1 STRESS The state of stress at any point in a loaded bo is defined completely in terms of the nine components of stress: σ xx,σ yy,σ zz,σ xy,σ yx,σ yz,σ zy,σ zx,andσ
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 informationMechanical Properties of Materials
Mechanical Properties of Materials Strains Material Model Stresses Learning objectives Understand the qualitative and quantitative description of mechanical properties of materials. Learn the logic of
More informationUNIVERSITY OF SASKATCHEWAN ME MECHANICS OF MATERIALS I FINAL EXAM DECEMBER 13, 2008 Professor A. Dolovich
UNIVERSITY OF SASKATCHEWAN ME 313.3 MECHANICS OF MATERIALS I FINAL EXAM DECEMBER 13, 2008 Professor A. Dolovich A CLOSED BOOK EXAMINATION TIME: 3 HOURS For Marker s Use Only LAST NAME (printed): FIRST
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 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 informationMechanics PhD Preliminary Spring 2017
Mechanics PhD Preliminary Spring 2017 1. (10 points) Consider a body Ω that is assembled by gluing together two separate bodies along a flat interface. The normal vector to the interface is given by n
More informationMEC-E8001 Finite Element Analysis, Exam (example) 2018
MEC-E8 inite Element Analysis Exam (example) 8. ind the transverse displacement wx ( ) of the structure consisting of one beam element and point forces and. he rotations of the endpoints are assumed to
More informationENERGY NORM A POSTERIORI ERROR ESTIMATES FOR MIXED FINITE ELEMENT METHODS
MATHEMATICS OF COMPUTATION Volume 75, Number 256, October 2006, Pages 1659 1674 S 0025-57180601872-2 Article electronically published on June 26, 2006 ENERGY NORM A POSTERIORI ERROR ESTIMATES FOR MIXED
More informationMacroscopic theory Rock as 'elastic continuum'
Elasticity and Seismic Waves Macroscopic theory Rock as 'elastic continuum' Elastic body is deformed in response to stress Two types of deformation: Change in volume and shape Equations of motion Wave
More informationKirchhoff Plates: Field Equations
20 Kirchhoff Plates: Field Equations AFEM Ch 20 Slide 1 Plate Structures A plate is a three dimensional bod characterized b Thinness: one of the plate dimensions, the thickness, is much smaller than the
More informationUnit 13 Review of Simple Beam Theory
MIT - 16.0 Fall, 00 Unit 13 Review of Simple Beam Theory Readings: Review Unified Engineering notes on Beam Theory BMP 3.8, 3.9, 3.10 T & G 10-15 Paul A. Lagace, Ph.D. Professor of Aeronautics & Astronautics
More informationNiiranen, Jarkko; Khakalo, Sergei; Balobanov, Viacheslav Strain gradient elasticity theories in lattice structure modelling
Powered by TCPDF (www.tcpdf.org) This is an electronic reprint of the original article. This reprint may differ from the original in pagination and typographic detail. Niiranen, Jarkko; Khakalo, Sergei;
More informationExample 3.7 Consider the undeformed configuration of a solid as shown in Figure 3.60.
162 3. The linear 3-D elasticity mathematical model The 3-D elasticity model is of great importance, since it is our highest order hierarchical model assuming linear elastic behavior. Therefore, it provides
More informationarxiv: v1 [math.na] 27 Jan 2016
Virtual Element Method for fourth order problems: L 2 estimates Claudia Chinosi a, L. Donatella Marini b arxiv:1601.07484v1 [math.na] 27 Jan 2016 a Dipartimento di Scienze e Innovazione Tecnologica, Università
More informationLourenço Beirão da Veiga, Jarkko Niiranen, Rolf Stenberg
D A family of C 0 finite elements for Kirchhoff plates II: Numerical results Lourenço Beirão da Veiga, Jarkko Niiranen, Rolf Stenberg Helsinki University of Technology, Institute of Mathematics Research
More informationCHAPTER THREE SYMMETRIC BENDING OF CIRCLE PLATES
CHAPTER THREE SYMMETRIC BENDING OF CIRCLE PLATES * Governing equations in beam and plate bending ** Solution by superposition 1.1 From Beam Bending to Plate Bending 1.2 Governing Equations For Symmetric
More informationConstitutive models. Constitutive model: determines P in terms of deformation
Constitutive models Constitutive model: determines P in terms of deformation Elastic material: P depends only on current F Hyperelastic material: work is independent of path strain energy density function
More informationPURE BENDING. If a simply supported beam carries two point loads of 10 kn as shown in the following figure, pure bending occurs at segment BC.
BENDING STRESS The effect of a bending moment applied to a cross-section of a beam is to induce a state of stress across that section. These stresses are known as bending stresses and they act normally
More informationLecture 8. Stress Strain in Multi-dimension
Lecture 8. Stress Strain in Multi-dimension Module. General Field Equations General Field Equations [] Equilibrium Equations in Elastic bodies xx x y z yx zx f x 0, etc [2] Kinematics xx u x x,etc. [3]
More informationMeasurement of deformation. Measurement of elastic force. Constitutive law. Finite element method
Deformable Bodies Deformation x p(x) Given a rest shape x and its deformed configuration p(x), how large is the internal restoring force f(p)? To answer this question, we need a way to measure deformation
More informationA HIGHER-ORDER BEAM THEORY FOR COMPOSITE BOX BEAMS
A HIGHER-ORDER BEAM THEORY FOR COMPOSITE BOX BEAMS A. Kroker, W. Becker TU Darmstadt, Department of Mechanical Engineering, Chair of Structural Mechanics Hochschulstr. 1, D-64289 Darmstadt, Germany kroker@mechanik.tu-darmstadt.de,
More informationChapter 12 Plate Bending Elements. Chapter 12 Plate Bending Elements
CIVL 7/8117 Chapter 12 - Plate Bending Elements 1/34 Chapter 12 Plate Bending Elements Learning Objectives To introduce basic concepts of plate bending. To derive a common plate bending element stiffness
More informationBilinear 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 information2 Introduction to mechanics
21 Motivation Thermodynamic bodies are being characterized by two competing opposite phenomena, energy and entropy which some researchers in thermodynamics would classify as cause and chance or determinancy
More informationMAE4700/5700 Finite Element Analysis for Mechanical and Aerospace Design
MAE4700/5700 Finite Element Analsis for Mechanical and Aerospace Design Cornell Universit, Fall 2009 Nicholas Zabaras Materials Process Design and Control Laborator Sible School of Mechanical and Aerospace
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 informationSymmetric Bending of Beams
Symmetric Bending of Beams beam is any long structural member on which loads act perpendicular to the longitudinal axis. Learning objectives Understand the theory, its limitations and its applications
More informationVariational principles in mechanics
CHAPTER Variational principles in mechanics.1 Linear Elasticity n D Figure.1: A domain and its boundary = D [. Consider a domain Ω R 3 with its boundary = D [ of normal n (see Figure.1). The problem of
More informationGEOMETRIC NONLINEAR ANALYSIS
GEOMETRIC NONLINEAR ANALYSIS The approach for solving problems with geometric nonlinearity is presented. The ESAComp solution relies on Elmer open-source computational tool [1] for multiphysics problems.
More informationLecture 8: Tissue Mechanics
Computational Biology Group (CoBi), D-BSSE, ETHZ Lecture 8: Tissue Mechanics Prof Dagmar Iber, PhD DPhil MSc Computational Biology 2015/16 7. Mai 2016 2 / 57 Contents 1 Introduction to Elastic Materials
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 informationChapter 3. Load and Stress Analysis
Chapter 3 Load and Stress Analysis 2 Shear Force and Bending Moments in Beams Internal shear force V & bending moment M must ensure equilibrium Fig. 3 2 Sign Conventions for Bending and Shear Fig. 3 3
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 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 information3 2 6 Solve the initial value problem u ( t) 3. a- If A has eigenvalues λ =, λ = 1 and corresponding eigenvectors 1
Math Problem a- If A has eigenvalues λ =, λ = 1 and corresponding eigenvectors 1 3 6 Solve the initial value problem u ( t) = Au( t) with u (0) =. 3 1 u 1 =, u 1 3 = b- True or false and why 1. if A is
More informationCOMPARISON OF PLATE MODELS FOR ANALYSIS OF LAMINATED COMPOSITES
COMPARISON OF PLATE MODELS FOR ANALYSIS OF LAMINATED COMPOSITES P. M. Mohite and C. S. Upadhyay** Department of Aerospace Engineering, IIT Kanpur 0806, INDIA, e-mail: mohite@iitk.ac.in Assistant Professor,
More informationToward a novel approach for damage identification and health monitoring of bridge structures
Toward a novel approach for damage identification and health monitoring of bridge structures Paolo Martino Calvi 1, Paolo Venini 1 1 Department of Structural Mechanics, University of Pavia, Italy E-mail:
More informationAircraft Structures Kirchhoff-Love Plates
University of Liège erospace & Mechanical Engineering ircraft Structures Kirchhoff-Love Plates Ludovic Noels Computational & Multiscale Mechanics of Materials CM3 http://www.ltas-cm3.ulg.ac.be/ Chemin
More informationBending of Simply Supported Isotropic and Composite Laminate Plates
Bending of Simply Supported Isotropic and Composite Laminate Plates Ernesto Gutierrez-Miravete 1 Isotropic Plates Consider simply a supported rectangular plate of isotropic material (length a, width b,
More informationPREPRINT 2010:23. A nonconforming rotated Q 1 approximation on tetrahedra PETER HANSBO
PREPRINT 2010:23 A nonconforming rotated Q 1 approximation on tetrahedra PETER HANSBO Department of Mathematical Sciences Division of Mathematics CHALMERS UNIVERSITY OF TECHNOLOGY UNIVERSITY OF GOTHENBURG
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 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 informationMIXED FINITE ELEMENT METHODS FOR PROBLEMS WITH ROBIN BOUNDARY CONDITIONS
MIXED FINITE ELEMENT METHODS FOR PROBLEMS WITH ROBIN BOUNDARY CONDITIONS JUHO KÖNNÖ, DOMINIK SCHÖTZAU, AND ROLF STENBERG Abstract. We derive new a-priori and a-posteriori error estimates for mixed nite
More informationSome New Elements for the Reissner Mindlin Plate Model
Boundary Value Problems for Partial Differential Equations and Applications, J.-L. Lions and C. Baiocchi, eds., Masson, 1993, pp. 287 292. Some New Elements for the Reissner Mindlin Plate Model Douglas
More informationSome Remarks on the Reissner Mindlin Plate Model
Some Remarks on the Reissner Mindlin Plate Model Alexandre L. Madureira LNCC Brazil Talk at the Workshop of Numerical Analysis of PDEs LNCC February 10, 2003 1 Outline The 3D Problem and its Modeling Full
More informationLECTURE # 0 BASIC NOTATIONS AND CONCEPTS IN THE THEORY OF PARTIAL DIFFERENTIAL EQUATIONS (PDES)
LECTURE # 0 BASIC NOTATIONS AND CONCEPTS IN THE THEORY OF PARTIAL DIFFERENTIAL EQUATIONS (PDES) RAYTCHO LAZAROV 1 Notations and Basic Functional Spaces Scalar function in R d, d 1 will be denoted by u,
More information2008 by authors and 2008 Springer Science+Business Media
Antti H. Niemi, Harri Hakula, and Juhani Pitkäranta. 28. Point load on a shell. In: Karl Kunisch, Günther Of, and Olaf Steinbach (editors). Numerical Mathematics and Advanced Applications. Proceedings
More informationContent. Department of Mathematics University of Oslo
Chapter: 1 MEK4560 The Finite Element Method in Solid Mechanics II (January 25, 2008) (E-post:torgeiru@math.uio.no) Page 1 of 14 Content 1 Introduction to MEK4560 3 1.1 Minimum Potential energy..............................
More informationUNCONVENTIONAL FINITE ELEMENT MODELS FOR NONLINEAR ANALYSIS OF BEAMS AND PLATES
UNCONVENTIONAL FINITE ELEMENT MODELS FOR NONLINEAR ANALYSIS OF BEAMS AND PLATES A Thesis by WOORAM KIM Submitted to the Office of Graduate Studies of Texas A&M University in partial fulfillment of the
More informationSEMM Mechanics PhD Preliminary Exam Spring Consider a two-dimensional rigid motion, whose displacement field is given by
SEMM Mechanics PhD Preliminary Exam Spring 2014 1. Consider a two-dimensional rigid motion, whose displacement field is given by u(x) = [cos(β)x 1 + sin(β)x 2 X 1 ]e 1 + [ sin(β)x 1 + cos(β)x 2 X 2 ]e
More informationCOMPRESSION AND BENDING STIFFNESS OF FIBER-REINFORCED ELASTOMERIC BEARINGS. Abstract. Introduction
COMPRESSION AND BENDING STIFFNESS OF FIBER-REINFORCED ELASTOMERIC BEARINGS Hsiang-Chuan Tsai, National Taiwan University of Science and Technology, Taipei, Taiwan James M. Kelly, University of California,
More informationReview of Strain Energy Methods and Introduction to Stiffness Matrix Methods of Structural Analysis
uke University epartment of Civil and Environmental Engineering CEE 42L. Matrix Structural Analysis Henri P. Gavin Fall, 22 Review of Strain Energy Methods and Introduction to Stiffness Matrix Methods
More informationThe Virtual Element Method: an introduction with focus on fluid flows
The Virtual Element Method: an introduction with focus on fluid flows L. Beirão da Veiga Dipartimento di Matematica e Applicazioni Università di Milano-Bicocca VEM people (or group representatives): P.
More informationA Mixed Nonconforming Finite Element for Linear Elasticity
A Mixed Nonconforming Finite Element for Linear Elasticity Zhiqiang Cai, 1 Xiu Ye 2 1 Department of Mathematics, Purdue University, West Lafayette, Indiana 47907-1395 2 Department of Mathematics and Statistics,
More informationFree vibration analysis of beams by using a third-order shear deformation theory
Sādhanā Vol. 32, Part 3, June 2007, pp. 167 179. Printed in India Free vibration analysis of beams by using a third-order shear deformation theory MESUT ŞİMŞEK and TURGUT KOCTÜRK Department of Civil Engineering,
More informationFind (u,p;λ), with u 0 and λ R, such that u + p = λu in Ω, (2.1) div u = 0 in Ω, u = 0 on Γ.
A POSTERIORI ESTIMATES FOR THE STOKES EIGENVALUE PROBLEM CARLO LOVADINA, MIKKO LYLY, AND ROLF STENBERG Abstract. We consider the Stokes eigenvalue problem. For the eigenvalues we derive both upper and
More informationAPPLICATION OF THE GALERKIN-VLASOV METHOD TO THE FLEXURAL ANALYSIS OF SIMPLY SUPPORTED RECTANGULAR KIRCHHOFF PLATES UNDER UNIFORM LOADS
Nigerian Journal of Technology (NIJOTECH) Vol. 35, No. 4, October 2016, pp. 732 738 Copyright Faculty of Engineering, University of Nigeria, Nsukka, Print ISSN: 0331-8443, Electronic ISSN: 2467-8821 www.nijotech.com
More informationTheories of Straight Beams
EVPM3ed02 2016/6/10 7:20 page 71 #25 This is a part of the revised chapter in the new edition of the tetbook Energy Principles and Variational Methods in pplied Mechanics, which will appear in 2017. These
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 information12. Stresses and Strains
12. Stresses and Strains Finite Element Method Differential Equation Weak Formulation Approximating Functions Weighted Residuals FEM - Formulation Classification of Problems Scalar Vector 1-D T(x) u(x)
More informationA short review of continuum mechanics
A short review of continuum mechanics Professor Anette M. Karlsson, Department of Mechanical ngineering, UD September, 006 This is a short and arbitrary review of continuum mechanics. Most of this material
More informationNomenclature. Length of the panel between the supports. Width of the panel between the supports/ width of the beam
omenclature a b c f h Length of the panel between the supports Width of the panel between the supports/ width of the beam Sandwich beam/ panel core thickness Thickness of the panel face sheet Sandwich
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: 5 Subdomain boundary nodes
Chapter: 5 MEK4560 The Finite Element Method in Solid Mechanics II (February 11, 2008) (E-post:torgeiru@math.uio.no) Page 1 of 19 5 Thick plates 3 5.1 ssumptions..................................... 3
More informationCHAPTER 5. Beam Theory
CHPTER 5. Beam Theory SangJoon Shin School of Mechanical and erospace Engineering Seoul National University ctive eroelasticity and Rotorcraft Lab. 5. The Euler-Bernoulli assumptions One of its dimensions
More information3D and Planar Constitutive Relations
3D and Planar Constitutive Relations A School on Mechanics of Fibre Reinforced Polymer Composites Knowledge Incubation for TEQIP Indian Institute of Technology Kanpur PM Mohite Department of Aerospace
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 informationINNOVATIVE FINITE ELEMENT METHODS FOR PLATES* DOUGLAS N. ARNOLD
INNOVATIVE FINITE ELEMENT METHODS FOR PLATES* DOUGLAS N. ARNOLD Abstract. Finite element methods for the Reissner Mindlin plate theory are discussed. Methods in which both the tranverse displacement and
More informationFINAL EXAMINATION. (CE130-2 Mechanics of Materials)
UNIVERSITY OF CLIFORNI, ERKELEY FLL SEMESTER 001 FINL EXMINTION (CE130- Mechanics of Materials) Problem 1: (15 points) pinned -bar structure is shown in Figure 1. There is an external force, W = 5000N,
More informationThe Rotating Inhomogeneous Elastic Cylinders of. Variable-Thickness and Density
Applied Mathematics & Information Sciences 23 2008, 237 257 An International Journal c 2008 Dixie W Publishing Corporation, U. S. A. The Rotating Inhomogeneous Elastic Cylinders of Variable-Thickness and
More informationNDT&E Methods: UT. VJ Technologies CAVITY INSPECTION. Nondestructive Testing & Evaluation TPU Lecture Course 2015/16.
CAVITY INSPECTION NDT&E Methods: UT VJ Technologies NDT&E Methods: UT 6. NDT&E: Introduction to Methods 6.1. Ultrasonic Testing: Basics of Elasto-Dynamics 6.2. Principles of Measurement 6.3. The Pulse-Echo
More informationA new approach for Kirchhoff-Love plates and shells
A new approach for Kirchhoff-Love plates and shells Walter Zulehner Institute of Computational Mathematics JKU Linz, Austria AANMPDE 10 October 2-6, 2017, Paleochora, Crete, Greece Walter Zulehner (JKU
More informationLecture 15 Strain and stress in beams
Spring, 2019 ME 323 Mechanics of Materials Lecture 15 Strain and stress in beams Reading assignment: 6.1 6.2 News: Instructor: Prof. Marcial Gonzalez Last modified: 1/6/19 9:42:38 PM Beam theory (@ ME
More informationCONSTITUTIVE RELATIONS FOR LINEAR ELASTIC SOLIDS
Chapter 9 CONSTITUTIV RLATIONS FOR LINAR LASTIC SOLIDS Figure 9.1: Hooke memorial window, St. Helen s, Bishopsgate, City of London 211 212 CHAPTR 9. CONSTITUTIV RLATIONS FOR LINAR LASTIC SOLIDS 9.1 Mechanical
More informationTable of Contents. Preface... 13
Table of Contents Preface... 13 Chapter 1. Vibrations of Continuous Elastic Solid Media... 17 1.1. Objective of the chapter... 17 1.2. Equations of motion and boundary conditions of continuous media...
More informationNonlinear Theory of Elasticity. Dr.-Ing. Martin Ruess
Nonlinear Theory of Elasticity Dr.-Ing. Martin Ruess geometry description Cartesian global coordinate system with base vectors of the Euclidian space orthonormal basis origin O point P domain of a deformable
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 informationA FULLY COUPLED MULTISCALE SHELL FORMULATION FOR THE MODELLING OF FIBRE REINFORCED LAMINATES
ECCM-6 TH EUROPEAN CONFERENCE ON COMPOSITE MATERIALS, Seville, Spain, 22-26 June 24 A FULLY COUPLED MULTISCALE SHELL FORMULATION FOR THE MODELLING OF FIBRE REINFORCED LAMINATES J. Främby, J. Brouzoulis,
More informationFLEXIBILITY METHOD FOR INDETERMINATE FRAMES
UNIT - I FLEXIBILITY METHOD FOR INDETERMINATE FRAMES 1. What is meant by indeterminate structures? Structures that do not satisfy the conditions of equilibrium are called indeterminate structure. These
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 informationComb resonator design (2)
Lecture 6: Comb resonator design () -Intro Intro. to Mechanics of Materials School of Electrical l Engineering i and Computer Science, Seoul National University Nano/Micro Systems & Controls Laboratory
More informationOn Static Instability and Estimates for Critical Velocities of Axially Moving Orthotropic Plates under Inhomogeneous Tension
Reports of the Department of Mathematical Information Technology Series B. Scientific Computing No. B. 8/212 On Static Instability and Estimates for Critical Velocities of Axially Moving Orthotropic Plates
More informationVYSOKÁ ŠKOLA BÁŇSKÁ TECHNICKÁ UNIVERZITA OSTRAVA
VYSOKÁ ŠKOLA BÁŇSKÁ TECHNICKÁ UNIVERZITA OSTRAVA FAKULTA METALURGIE A MATERIÁLOVÉHO INŽENÝRSTVÍ APPLIED MECHANICS Study Support Leo Václavek Ostrava 2015 Title:Applied Mechanics Code: Author: doc. Ing.
More informationChapter 5 Elastic Strain, Deflection, and Stability 1. Elastic Stress-Strain Relationship
Chapter 5 Elastic Strain, Deflection, and Stability Elastic Stress-Strain Relationship A stress in the x-direction causes a strain in the x-direction by σ x also causes a strain in the y-direction & z-direction
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 informationarxiv: v1 [math.na] 27 Jan 2016
Virtual Element Method for fourth order problems: L 2 estimates Claudia Chinosi a, L. Donatella Marini b arxiv:1601.07484v1 [math.na] 27 Jan 2016 a Dipartimento di Scienze e Innovazione Tecnologica, Università
More informationFinite Elements for Elastic Shell Models in
Elastic s in Advisor: Matthias Heinkenschloss Computational and Applied Mathematics Rice University 13 April 2007 Outline Elasticity in Differential Geometry of Shell Geometry and Equations The Plate Model
More informationFundamentals of Linear Elasticity
Fundamentals of Linear Elasticity Introductory Course on Multiphysics Modelling TOMASZ G. ZIELIŃSKI bluebox.ippt.pan.pl/ tzielins/ Institute of Fundamental Technological Research of the Polish Academy
More informationPEAT SEISMOLOGY Lecture 2: Continuum mechanics
PEAT8002 - SEISMOLOGY Lecture 2: Continuum mechanics Nick Rawlinson Research School of Earth Sciences Australian National University Strain Strain is the formal description of the change in shape of a
More information7. Hierarchical modeling examples
7. Hierarchical modeling examples The objective of this chapter is to apply the hierarchical modeling approach discussed in Chapter 1 to three selected problems using the mathematical models studied in
More informationUsing MATLAB and. Abaqus. Finite Element Analysis. Introduction to. Amar Khennane. Taylor & Francis Croup. Taylor & Francis Croup,
Introduction to Finite Element Analysis Using MATLAB and Abaqus Amar Khennane Taylor & Francis Croup Boca Raton London New York CRC Press is an imprint of the Taylor & Francis Croup, an informa business
More informationStructural Dynamics Lecture Eleven: Dynamic Response of MDOF Systems: (Chapter 11) By: H. Ahmadian
Structural Dynamics Lecture Eleven: Dynamic Response of MDOF Systems: (Chapter 11) By: H. Ahmadian ahmadian@iust.ac.ir Dynamic Response of MDOF Systems: Mode-Superposition Method Mode-Superposition Method:
More informationElements of Continuum Elasticity. David M. Parks Mechanics and Materials II February 25, 2004
Elements of Continuum Elasticity David M. Parks Mechanics and Materials II 2.002 February 25, 2004 Solid Mechanics in 3 Dimensions: stress/equilibrium, strain/displacement, and intro to linear elastic
More informationMechanics of Earthquakes and Faulting
Mechanics of Earthquakes and Faulting www.geosc.psu.edu/courses/geosc508 Overview Milestones in continuum mechanics Concepts of modulus and stiffness. Stress-strain relations Elasticity Surface and body
More information