Finite Element Solution Algorithm for Nonlinear Elasticity Problems by Domain Decomposition Method
|
|
- Derek Sherman
- 5 years ago
- Views:
Transcription
1 Fnte Element Soluton Algorthm for Nonlnear Elastcty Problems by Doman Decomposton Method Bedřch Sousedík Department of Mathematcs, Faculty of Cvl Engneerng, Czech Techncal Unversty n Prague, Thákurova 7, Prague 6 sousedk@matfsvcvutcz Abstract The purpose of the present work s to gve a bref descrpton of the fnte elastcty and of ts approxmaton va fnte element method We formulate the problem for the case of compressble elastcty Weak formulaton allows to use any sotropc hyperelastc materal model that satsfes polyconvexty assumptons Dscretzaton usng FEM leads to systems of non-lnear equatons Such systems of equatons can be solved by Newton s method and ts modfcatons In each teraton of Newton s method a lnearzed system of equatons has to be solved We propose to use BDDC precondtonng teratve method for the soluton of lnear system of equatons 1 Introducton The man object of the fnte three-dmensonal elastcty s to predct changes n the geometry of sold bodes The startng pont of the classcal theory of lnear elastcty s the concept of small strans: the deformaton of structures under workng loads are not detectable by human eye In contrast, many modern stuatons nvolve large deformatons The nonlnear behavor of polymers and synthetc rubbers are such examples Applcatons n bomechancs are even more crtcal because the most of vtal organs such as eye, heart trachea or vocal apparatus fulfll ther functon only because of ther large deformatons In ths framework the concept of fnte elastcty covers the smplest case where nternal forces (stresses) depend only on the present deformaton of the body and not on the hstory In the paper we show the fnte element approxmaton and strategy of soluton of ths nonlnear and vsble stress-stran relatonshp 1
2 2 Formulaton of elastcty problem Let us descrbe the deformaton of a body by ϕ : ϕ, and the occuped doman we decompose n an nteror and boundary as ϕ = ϕ Γ ϕ, (e Γ ϕ = ϕ ) The classc formulaton of equlbrum equaton we wrte as 21 Weak formulaton T ϕ j x ϕ j + f ϕ = 0 (1) Multplcaton by test functon v ϕ, ntegraton over the whole doman ϕ and applcaton of Green s theorem gves T ϕ j nϕ j vϕ daϕ T ϕ v ϕ j dx ϕ + f ϕ vϕ dx ϕ = 0 (2) ϕ ϕ ϕ x ϕ j On the parts of boundary Γ ϕ = Γ ϕ v Γ ϕ τ we prescrbe followng boundary condtons T ϕ j nϕ j = g ϕ on Γ ϕ τ (3) v ϕ = 0 on Γ ϕ v (4) After some rearrangements we obtan the weak formulaton of equlbrum equatons of a body after deformaton T ϕ v ϕ j ϕ x ϕ j dx ϕ = f ϕ vϕ dxϕ + ϕ Γ ϕ τ g ϕ vϕ daϕ (5) Unfortunatelly, n fnte elastcty, ϕ s unknown and may be very dfferent from the known reference confguraton Therefore, t s more convenent to rewrte the equlbrum equatons on, usng the formula for changes of varables n multple ntegrals Dong ths, usng Pola transform and consderng that x = ϕ 1 (x ϕ ) we fnally obtan the followng weak form of equlbrum equaton n the reference confguraton v T k dx = f v dx + g v da v V, (6) x k Γ τ 2
3 Consderng the followng consttutve relaton for compressble materal we get T j (u) = Ŵ F j (x,f(u)), (7) Ŵ (x,f(u)) v dx = f v dx + g v da v V (8) F j x j Γ τ 3 Consttutve relatons The smplest law uses a quadratc sotropc functon of the Green stran tensor E = 1 2 (C I) So called St Venant materal unfortunately can reach nfnte compresson rates wth fnte energy and do not satsfy the polyconvexty assumptons used n the exstence theory For these reason we do not use ths materal model here and suggest to use polyconvex functons gven n terms of nvarants The smplest example of such materals s the neo-hookean materal W = C 10 (I 1 3) (9) If we add a lnear term, we get well-known Mooney-Rvln materal W = C 10 (I 1 3) + C 01 (I 2 3) (10) Ths energy functon was further generalzed nto the thrd order polynomal n nvarants I 1,I 2 whch fts well to numerous expermental data W = C 10 (I 1 3) + C 01 (I 2 3) + C 20 (I 1 3) 2 + C 02 (I 2 3) 2 + C 11 (I 1 3)(I 2 3) + C 30 (I 1 3) 3 (11) Theorem (Rvln-Erksen representaton theorem): For any sotropc hyperelastc materal, the elastc potental W satsfes: W(x,F) = W(x,I 1 (E),I 2 (E),I 3 (E)) (12) Invarants and ther dervatves are gven n the followng table: I 1 = tre I 1 E j = δ j I 2 = 1 2 tre2 = 1 2 E je j I 2 E j = E j (13) I 3 = 1 3 tre3 = 1 3 E je jk E k I 3 E j = E k E kj 3
4 4 Numercal soluton technque Equlbrum equatons are generally nonlnear n the dsplacements u We would lke to fnd the feld of dsplacements u so that F(u) = 0 usng Newton s method n the followng way: n the (k+1)-th teraton of Newton s method we are lookng for the feld of dsplacements u k+1 The feld of dsplacements from k-th teraton u k s known and we must fnd ncrement h R n satsfyng the equaton DF(u k ) h = F(u k ), (14) then we add the ncrement h to the prevous teraton so that Naturally, n our case, F(u) = u k+1 = u k + h (15) Ŵ (x,f(u)) w dx fw dx gw da (16) F j x j Γ τ and the Fréchet dervatve of F(u) s D v F(u) = ( 2 Ŵ(x,F(u)) v k F j F kl x l 5 Fnte Element Formulaton ) w x j dx, (17) Frst we use the equvalence relaton W = Ŵ(F) = W(C) followng from defnton C j = F k F kj = δ j + 2E j to transform F(u) and DF(u) n order to use nternal energy functons n terms of nvarants and stran tensors E j F(u) = [( W (δ k + u )] k wk dx fw dx gw da (18) E j x x j Γ τ and usng the second Pola-Krchhoff stress tensor S j = 2 W C j we get D v F(u) = { 2 W E j E kl [( δ r + u r x 4 ) vr x j ][( δ sk + u s x k ) vs x l ] } v k w k + S j dx(19) x x j
5 Takng w = (0,N t,0) and v = h = (h x,h y,h z ) where h x = N h u=1 h xun u h y = N h u=1 h yun u h z = N h u=1 h zun u (20) we get the components of the stffness matrx and rght hand sde vectors nto Newton s method Fnally, the (k+1)-th teraton of Newton s method conssts of two steps: () Frst, we solve the followng system of equatons A k h = b k, (21) where the components of the matrx A and vectors h, b are defned as A = a xx a xy a xz a yx a yy a yz a zx a zy a zz (22) b = b x b y b z h = h x h y h z (23) () Next, we update the feld of dsplacements u k+1 = u k + h (24) 6 Formulaton of BDDC method Clearly, n each step of Newton s teratons we solve a lnear system of equaton Here, we descrbe the BDDC precondtonng method Ths method s closely related to FETI-DP method that s also brefly descrbed Let us decompose the doman nto the nonoverlappng set of substructures (subdomans), where = 1,N and N means the number of substructures Let K be the stffness matrx and v the vector of degrees of freedom (dofs) for substructure We want to solve the problem n decomposed form 1 2 vt Kv v T f mn (25) 5
6 where K 1 0 K = 0 K N, v = v 1 v N, f = subject to contnuty dofs between substructures Parttonng the dofs n each subdoman nto nternal and nterface (boundary) K = [ K K b T K b K bb ] [ v, v = v b ], f = and elmnatng the nteror dofs we obtan the problem reduced to nterfaces 1 2 wt Sw w T g mn, S = dag(s ), S = K bb [ f f b f 1 f N ], K b T K 1 K b, agan subject to contnuty of dofs between substructures In BDD type methods, the contnuty of dofs between substructures s enforced by mposng common values on substructures nterfaces: w = Ru for some u, where R = and R s the operator of restrcton of global dofs on the nterfaces to substructure In FETI type methods, contnuty of dofs between substructures s enforced by the constrant Bw = 0, where the entres of B are typcally 0, ±1 By constructon, we have R 1 R N R R T = I, range R = nullb A BDDC or FETI-DP method s specfed by the choce of coarse dofs and the choce of weghts for ntersubdoman averagng To defne the coarse problem for BDDC, choose a matrx Q T P that selects coarse dofs u c from global nterface dofs u, eg as values at corners or averages on sdes: u c = Q T P u We defne W as the space of all vectors of substructure nterface dofs that are contnuous between substructures, W = {w W : u c : Cw = R c u c } = {w W : Cw range R c } (26) 6
7 where C 1 0 C = R c Q T P RT, C = 0 C N, R c = R c1 R cn, (27) and the matrx R c restrcts the vector all coarse dof values nto a vector of coarse dof values that can be nonzero on substructure The dual approach n FETI-DP s to construct Q D such that W = { w W : Q T D Bw = 0} (28) In BDDC, the ntersubdoman averagng s defned by the matrces that form a decomposton of unty, D P = dag (D P ) R T D P R = I The correspondng dual matrces n FETI-DP are are B D = [D D1 B 1, D DN B N ], where the dual weghts D D are defned so that B T D B + RRT D P = I (29) The BDDC method s then the method of conjugate gradents for the assembled system Au = R T g wth the system matrx and the precondtoner P defned by A = R T SR Pr = R T D P (Ψu c + z), where u c s the soluton of the coarse problem and z s the soluton of Ψ T SΨu c = Ψ T D T P Rr Sz + C T µ = DP TRr, Cz = 0 whch s a collecton of ndependent substructure problems The coarse bass functons Ψ are defned by energy mnmzaton, [ ][ ] [ ] S C T Ψ 0 = C 0 Λ R c 7
8 7 Concluson In the paper we showed fnte element approxmaton of the non-lnear threedmensonal fnte elastcty problem for compressble materal model We also descrbed the strategy of soluton of the non-lnear system of equatons arsed The lnearzed system of equatons wll be solved by an teratve solver of conjugate gradent type wth BDDC precondtonng proposed and studed n [1], [3] Currently we are developng and testng ths fnte element code n programmng language FORTRAN We wll also contnue n developng of adaptve strateges for the selecton of constrants studed n the paper [4] The whole code wll be fnally parallelzed usng MPI lbrary Acknowledgement The work was supported by grant GA ČR 106/05/2731 and by the Program Informaton Socety under project IET References [1] Dohrmann, CR: A precondtoner for substructurng based on constraned energy mnmzaton, SIAM J Sc Comput, 25(1): , 2003 [2] Le Tallec, P: Numercal Methods for Nonlnear Three-dmensonal Elastcty, n Handbook of Numercal Analyss, Vol III, (P G Carlet and J L Lons eds) North-Holland, 1994 [3] Mandel, J, Dohrmann, CR: Convergence of balancng doman decomposton by constrants and energy mnmzaton, Numer Lnear Algebra Appl 10: , 2003 [4] Mandel, J, Sousedík, B: Adaptve Coarse Space Selecton n BDDC and FETI-DP Iteratve Substructurng Methods: Optmal Face Degrees of Freedom, 16th Internatonal Doman Decomposton Conference, New York, January 2005, Lecture Notes n Computatonal Scence, submtted [5] Sousedík, B, Burda, P: Fnte Element Formulaton of the Threedmensonal Non-lnear Elastcty Problem, Semnar n Appled Mathematcs, Czech Techncal Unversty n Prague, Prague, Aprl
Adaptive Coarse Space Selection in the BDDC and the FETI-DP Iterative Substructuring Methods: Optimal Face Degrees of Freedom
Adaptve Coarse Space Selecton n the BDDC and the FETI-DP Iteratve Substructurng Methods: Optmal Face Degrees of Freedom Jan Mandel and Bedřch Sousedík 2 Department of Mathematcs, Unversty of Colorado at
More informationFinite Element Modelling of truss/cable structures
Pet Schreurs Endhoven Unversty of echnology Department of Mechancal Engneerng Materals echnology November 3, 214 Fnte Element Modellng of truss/cable structures 1 Fnte Element Analyss of prestressed structures
More informationIn this section is given an overview of the common elasticity models.
Secton 4.1 4.1 Elastc Solds In ths secton s gven an overvew of the common elastcty models. 4.1.1 The Lnear Elastc Sold The classcal Lnear Elastc model, or Hooean model, has the followng lnear relatonshp
More informationLecture 21: Numerical methods for pricing American type derivatives
Lecture 21: Numercal methods for prcng Amercan type dervatves Xaoguang Wang STAT 598W Aprl 10th, 2014 (STAT 598W) Lecture 21 1 / 26 Outlne 1 Fnte Dfference Method Explct Method Penalty Method (STAT 598W)
More informationELASTIC WAVE PROPAGATION IN A CONTINUOUS MEDIUM
ELASTIC WAVE PROPAGATION IN A CONTINUOUS MEDIUM An elastc wave s a deformaton of the body that travels throughout the body n all drectons. We can examne the deformaton over a perod of tme by fxng our look
More informationMMA and GCMMA two methods for nonlinear optimization
MMA and GCMMA two methods for nonlnear optmzaton Krster Svanberg Optmzaton and Systems Theory, KTH, Stockholm, Sweden. krlle@math.kth.se Ths note descrbes the algorthms used n the author s 2007 mplementatons
More informationGroup Analysis of Ordinary Differential Equations of the Order n>2
Symmetry n Nonlnear Mathematcal Physcs 997, V., 64 7. Group Analyss of Ordnary Dfferental Equatons of the Order n> L.M. BERKOVICH and S.Y. POPOV Samara State Unversty, 4430, Samara, Russa E-mal: berk@nfo.ssu.samara.ru
More informationInner Product. Euclidean Space. Orthonormal Basis. Orthogonal
Inner Product Defnton 1 () A Eucldean space s a fnte-dmensonal vector space over the reals R, wth an nner product,. Defnton 2 (Inner Product) An nner product, on a real vector space X s a symmetrc, blnear,
More informationPerron Vectors of an Irreducible Nonnegative Interval Matrix
Perron Vectors of an Irreducble Nonnegatve Interval Matrx Jr Rohn August 4 2005 Abstract As s well known an rreducble nonnegatve matrx possesses a unquely determned Perron vector. As the man result of
More informationSome modelling aspects for the Matlab implementation of MMA
Some modellng aspects for the Matlab mplementaton of MMA Krster Svanberg krlle@math.kth.se Optmzaton and Systems Theory Department of Mathematcs KTH, SE 10044 Stockholm September 2004 1. Consdered optmzaton
More information1 Matrix representations of canonical matrices
1 Matrx representatons of canoncal matrces 2-d rotaton around the orgn: ( ) cos θ sn θ R 0 = sn θ cos θ 3-d rotaton around the x-axs: R x = 1 0 0 0 cos θ sn θ 0 sn θ cos θ 3-d rotaton around the y-axs:
More informationMath 217 Fall 2013 Homework 2 Solutions
Math 17 Fall 013 Homework Solutons Due Thursday Sept. 6, 013 5pm Ths homework conssts of 6 problems of 5 ponts each. The total s 30. You need to fully justfy your answer prove that your functon ndeed has
More informationU.C. Berkeley CS294: Beyond Worst-Case Analysis Luca Trevisan September 5, 2017
U.C. Berkeley CS94: Beyond Worst-Case Analyss Handout 4s Luca Trevsan September 5, 07 Summary of Lecture 4 In whch we ntroduce semdefnte programmng and apply t to Max Cut. Semdefnte Programmng Recall that
More informationLecture Note 3. Eshelby s Inclusion II
ME340B Elastcty of Mcroscopc Structures Stanford Unversty Wnter 004 Lecture Note 3. Eshelby s Incluson II Chrs Wenberger and We Ca c All rghts reserved January 6, 004 Contents 1 Incluson energy n an nfnte
More informationSalmon: Lectures on partial differential equations. Consider the general linear, second-order PDE in the form. ,x 2
Salmon: Lectures on partal dfferental equatons 5. Classfcaton of second-order equatons There are general methods for classfyng hgher-order partal dfferental equatons. One s very general (applyng even to
More informationCOMPOSITE BEAM WITH WEAK SHEAR CONNECTION SUBJECTED TO THERMAL LOAD
COMPOSITE BEAM WITH WEAK SHEAR CONNECTION SUBJECTED TO THERMAL LOAD Ákos Jósef Lengyel, István Ecsed Assstant Lecturer, Professor of Mechancs, Insttute of Appled Mechancs, Unversty of Mskolc, Mskolc-Egyetemváros,
More informationPHYS 705: Classical Mechanics. Calculus of Variations II
1 PHYS 705: Classcal Mechancs Calculus of Varatons II 2 Calculus of Varatons: Generalzaton (no constrant yet) Suppose now that F depends on several dependent varables : We need to fnd such that has a statonary
More informationSIMULATION OF WAVE PROPAGATION IN AN HETEROGENEOUS ELASTIC ROD
SIMUATION OF WAVE POPAGATION IN AN HETEOGENEOUS EASTIC OD ogéro M Saldanha da Gama Unversdade do Estado do o de Janero ua Sào Francsco Xaver 54, sala 5 A 559-9, o de Janero, Brasl e-mal: rsgama@domancombr
More informationThe Two-scale Finite Element Errors Analysis for One Class of Thermoelastic Problem in Periodic Composites
7 Asa-Pacfc Engneerng Technology Conference (APETC 7) ISBN: 978--6595-443- The Two-scale Fnte Element Errors Analyss for One Class of Thermoelastc Problem n Perodc Compostes Xaoun Deng Mngxang Deng ABSTRACT
More informationLifetime prediction of EP and NBR rubber seal by thermos-viscoelastic model
ECCMR, Prague, Czech Republc; September 3 th, 2015 Lfetme predcton of EP and NBR rubber seal by thermos-vscoelastc model Kotaro KOBAYASHI, Takahro ISOZAKI, Akhro MATSUDA Unversty of Tsukuba, Japan Yoshnobu
More informationLectures - Week 4 Matrix norms, Conditioning, Vector Spaces, Linear Independence, Spanning sets and Basis, Null space and Range of a Matrix
Lectures - Week 4 Matrx norms, Condtonng, Vector Spaces, Lnear Independence, Spannng sets and Bass, Null space and Range of a Matrx Matrx Norms Now we turn to assocatng a number to each matrx. We could
More informationA New Refinement of Jacobi Method for Solution of Linear System Equations AX=b
Int J Contemp Math Scences, Vol 3, 28, no 17, 819-827 A New Refnement of Jacob Method for Soluton of Lnear System Equatons AX=b F Naem Dafchah Department of Mathematcs, Faculty of Scences Unversty of Gulan,
More informationThe Finite Element Method
The Fnte Element Method GENERAL INTRODUCTION Read: Chapters 1 and 2 CONTENTS Engneerng and analyss Smulaton of a physcal process Examples mathematcal model development Approxmate solutons and methods of
More informationLecture 5 Decoding Binary BCH Codes
Lecture 5 Decodng Bnary BCH Codes In ths class, we wll ntroduce dfferent methods for decodng BCH codes 51 Decodng the [15, 7, 5] 2 -BCH Code Consder the [15, 7, 5] 2 -code C we ntroduced n the last lecture
More informationAPPENDIX A Some Linear Algebra
APPENDIX A Some Lnear Algebra The collecton of m, n matrces A.1 Matrces a 1,1,..., a 1,n A = a m,1,..., a m,n wth real elements a,j s denoted by R m,n. If n = 1 then A s called a column vector. Smlarly,
More informationOn a direct solver for linear least squares problems
ISSN 2066-6594 Ann. Acad. Rom. Sc. Ser. Math. Appl. Vol. 8, No. 2/2016 On a drect solver for lnear least squares problems Constantn Popa Abstract The Null Space (NS) algorthm s a drect solver for lnear
More informationCHAPTER 5 NUMERICAL EVALUATION OF DYNAMIC RESPONSE
CHAPTER 5 NUMERICAL EVALUATION OF DYNAMIC RESPONSE Analytcal soluton s usually not possble when exctaton vares arbtrarly wth tme or f the system s nonlnear. Such problems can be solved by numercal tmesteppng
More informationNUMERICAL RESULTS QUALITY IN DEPENDENCE ON ABAQUS PLANE STRESS ELEMENTS TYPE IN BIG DISPLACEMENTS COMPRESSION TEST
Appled Computer Scence, vol. 13, no. 4, pp. 56 64 do: 10.23743/acs-2017-29 Submtted: 2017-10-30 Revsed: 2017-11-15 Accepted: 2017-12-06 Abaqus Fnte Elements, Plane Stress, Orthotropc Materal Bartosz KAWECKI
More informationA Hybrid Variational Iteration Method for Blasius Equation
Avalable at http://pvamu.edu/aam Appl. Appl. Math. ISSN: 1932-9466 Vol. 10, Issue 1 (June 2015), pp. 223-229 Applcatons and Appled Mathematcs: An Internatonal Journal (AAM) A Hybrd Varatonal Iteraton Method
More information4DVAR, according to the name, is a four-dimensional variational method.
4D-Varatonal Data Assmlaton (4D-Var) 4DVAR, accordng to the name, s a four-dmensonal varatonal method. 4D-Var s actually a drect generalzaton of 3D-Var to handle observatons that are dstrbuted n tme. The
More informationLecture 12: Discrete Laplacian
Lecture 12: Dscrete Laplacan Scrbe: Tanye Lu Our goal s to come up wth a dscrete verson of Laplacan operator for trangulated surfaces, so that we can use t n practce to solve related problems We are mostly
More informationON A DETERMINATION OF THE INITIAL FUNCTIONS FROM THE OBSERVED VALUES OF THE BOUNDARY FUNCTIONS FOR THE SECOND-ORDER HYPERBOLIC EQUATION
Advanced Mathematcal Models & Applcatons Vol.3, No.3, 2018, pp.215-222 ON A DETERMINATION OF THE INITIAL FUNCTIONS FROM THE OBSERVED VALUES OF THE BOUNDARY FUNCTIONS FOR THE SECOND-ORDER HYPERBOLIC EUATION
More informationCOMPARISON OF SOME RELIABILITY CHARACTERISTICS BETWEEN REDUNDANT SYSTEMS REQUIRING SUPPORTING UNITS FOR THEIR OPERATIONS
Avalable onlne at http://sck.org J. Math. Comput. Sc. 3 (3), No., 6-3 ISSN: 97-537 COMPARISON OF SOME RELIABILITY CHARACTERISTICS BETWEEN REDUNDANT SYSTEMS REQUIRING SUPPORTING UNITS FOR THEIR OPERATIONS
More informationNumerical Heat and Mass Transfer
Master degree n Mechancal Engneerng Numercal Heat and Mass Transfer 06-Fnte-Dfference Method (One-dmensonal, steady state heat conducton) Fausto Arpno f.arpno@uncas.t Introducton Why we use models and
More informationInexact Newton Methods for Inverse Eigenvalue Problems
Inexact Newton Methods for Inverse Egenvalue Problems Zheng-jan Ba Abstract In ths paper, we survey some of the latest development n usng nexact Newton-lke methods for solvng nverse egenvalue problems.
More informationEVALUATION OF THE VISCO-ELASTIC PROPERTIES IN ASPHALT RUBBER AND CONVENTIONAL MIXES
EVALUATION OF THE VISCO-ELASTIC PROPERTIES IN ASPHALT RUBBER AND CONVENTIONAL MIXES Manuel J. C. Mnhoto Polytechnc Insttute of Bragança, Bragança, Portugal E-mal: mnhoto@pb.pt Paulo A. A. Perera and Jorge
More informationCollege of Computer & Information Science Fall 2009 Northeastern University 20 October 2009
College of Computer & Informaton Scence Fall 2009 Northeastern Unversty 20 October 2009 CS7880: Algorthmc Power Tools Scrbe: Jan Wen and Laura Poplawsk Lecture Outlne: Prmal-dual schema Network Desgn:
More informationTR A BDDC ALGORITHM FOR PROBLEMS WITH MORTAR DISCRETIZATION. September 4, 2005
TR2005-873 A BDDC ALGORITHM FOR PROBLEMS WITH MORTAR DISCRETIZATION HYEA HYUN KIM, MAKSYMILIAN DRYJA, AND OLOF B WIDLUND September 4, 2005 Abstract A BDDC balancng doman decomposton by constrants algorthm
More informationLecture 10 Support Vector Machines II
Lecture 10 Support Vector Machnes II 22 February 2016 Taylor B. Arnold Yale Statstcs STAT 365/665 1/28 Notes: Problem 3 s posted and due ths upcomng Frday There was an early bug n the fake-test data; fxed
More informationThe Second Anti-Mathima on Game Theory
The Second Ant-Mathma on Game Theory Ath. Kehagas December 1 2006 1 Introducton In ths note we wll examne the noton of game equlbrum for three types of games 1. 2-player 2-acton zero-sum games 2. 2-player
More informationTransfer Functions. Convenient representation of a linear, dynamic model. A transfer function (TF) relates one input and one output: ( ) system
Transfer Functons Convenent representaton of a lnear, dynamc model. A transfer functon (TF) relates one nput and one output: x t X s y t system Y s The followng termnology s used: x y nput output forcng
More informationLeast squares cubic splines without B-splines S.K. Lucas
Least squares cubc splnes wthout B-splnes S.K. Lucas School of Mathematcs and Statstcs, Unversty of South Australa, Mawson Lakes SA 595 e-mal: stephen.lucas@unsa.edu.au Submtted to the Gazette of the Australan
More information5 The Rational Canonical Form
5 The Ratonal Canoncal Form Here p s a monc rreducble factor of the mnmum polynomal m T and s not necessarly of degree one Let F p denote the feld constructed earler n the course, consstng of all matrces
More information2. PROBLEM STATEMENT AND SOLUTION STRATEGIES. L q. Suppose that we have a structure with known geometry (b, h, and L) and material properties (EA).
. PROBEM STATEMENT AND SOUTION STRATEGIES Problem statement P, Q h ρ ρ o EA, N b b Suppose that we have a structure wth known geometry (b, h, and ) and materal propertes (EA). Gven load (P), determne the
More informationPlate Theories for Classical and Laminated plates Weak Formulation and Element Calculations
Plate heores for Classcal and Lamnated plates Weak Formulaton and Element Calculatons PM Mohte Department of Aerospace Engneerng Indan Insttute of echnolog Kanpur EQIP School on Computatonal Methods n
More informationVARIATION OF CONSTANT SUM CONSTRAINT FOR INTEGER MODEL WITH NON UNIFORM VARIABLES
VARIATION OF CONSTANT SUM CONSTRAINT FOR INTEGER MODEL WITH NON UNIFORM VARIABLES BÂRZĂ, Slvu Faculty of Mathematcs-Informatcs Spru Haret Unversty barza_slvu@yahoo.com Abstract Ths paper wants to contnue
More informationA new Approach for Solving Linear Ordinary Differential Equations
, ISSN 974-57X (Onlne), ISSN 974-5718 (Prnt), Vol. ; Issue No. 1; Year 14, Copyrght 13-14 by CESER PUBLICATIONS A new Approach for Solvng Lnear Ordnary Dfferental Equatons Fawz Abdelwahd Department of
More informationPHYS 705: Classical Mechanics. Newtonian Mechanics
1 PHYS 705: Classcal Mechancs Newtonan Mechancs Quck Revew of Newtonan Mechancs Basc Descrpton: -An dealzed pont partcle or a system of pont partcles n an nertal reference frame [Rgd bodes (ch. 5 later)]
More informationReport on Image warping
Report on Image warpng Xuan Ne, Dec. 20, 2004 Ths document summarzed the algorthms of our mage warpng soluton for further study, and there s a detaled descrpton about the mplementaton of these algorthms.
More informationProjective change between two Special (α, β)- Finsler Metrics
Internatonal Journal of Trend n Research and Development, Volume 2(6), ISSN 2394-9333 www.jtrd.com Projectve change between two Specal (, β)- Fnsler Metrcs Gayathr.K 1 and Narasmhamurthy.S.K 2 1 Assstant
More informationThe Order Relation and Trace Inequalities for. Hermitian Operators
Internatonal Mathematcal Forum, Vol 3, 08, no, 507-57 HIKARI Ltd, wwwm-hkarcom https://doorg/0988/mf088055 The Order Relaton and Trace Inequaltes for Hermtan Operators Y Huang School of Informaton Scence
More informationThe Exact Formulation of the Inverse of the Tridiagonal Matrix for Solving the 1D Poisson Equation with the Finite Difference Method
Journal of Electromagnetc Analyss and Applcatons, 04, 6, 0-08 Publshed Onlne September 04 n ScRes. http://www.scrp.org/journal/jemaa http://dx.do.org/0.46/jemaa.04.6000 The Exact Formulaton of the Inverse
More informationModule 3: Element Properties Lecture 1: Natural Coordinates
Module 3: Element Propertes Lecture : Natural Coordnates Natural coordnate system s bascally a local coordnate system whch allows the specfcaton of a pont wthn the element by a set of dmensonless numbers
More informationFixed point method and its improvement for the system of Volterra-Fredholm integral equations of the second kind
MATEMATIKA, 217, Volume 33, Number 2, 191 26 c Penerbt UTM Press. All rghts reserved Fxed pont method and ts mprovement for the system of Volterra-Fredholm ntegral equatons of the second knd 1 Talaat I.
More informationVisco-Rubber Elastic Model for Pressure Sensitive Adhesive
Vsco-Rubber Elastc Model for Pressure Senstve Adhesve Kazuhsa Maeda, Shgenobu Okazawa, Koj Nshgch and Takash Iwamoto Abstract A materal model to descrbe large deformaton of pressure senstve adhesve (PSA
More informationAPPENDIX F A DISPLACEMENT-BASED BEAM ELEMENT WITH SHEAR DEFORMATIONS. Never use a Cubic Function Approximation for a Non-Prismatic Beam
APPENDIX F A DISPACEMENT-BASED BEAM EEMENT WITH SHEAR DEFORMATIONS Never use a Cubc Functon Approxmaton for a Non-Prsmatc Beam F. INTRODUCTION { XE "Shearng Deformatons" }In ths appendx a unque development
More informationA particle in a state of uniform motion remain in that state of motion unless acted upon by external force.
The fundamental prncples of classcal mechancs were lad down by Galleo and Newton n the 16th and 17th centures. In 1686, Newton wrote the Prncpa where he gave us three laws of moton, one law of gravty,
More informationBézier curves. Michael S. Floater. September 10, These notes provide an introduction to Bézier curves. i=0
Bézer curves Mchael S. Floater September 1, 215 These notes provde an ntroducton to Bézer curves. 1 Bernsten polynomals Recall that a real polynomal of a real varable x R, wth degree n, s a functon of
More informationThe Finite Element Method: A Short Introduction
Te Fnte Element Metod: A Sort ntroducton Wat s FEM? Te Fnte Element Metod (FEM) ntroduced by engneers n late 50 s and 60 s s a numercal tecnque for solvng problems wc are descrbed by Ordnary Dfferental
More informationIntegrals and Invariants of Euler-Lagrange Equations
Lecture 16 Integrals and Invarants of Euler-Lagrange Equatons ME 256 at the Indan Insttute of Scence, Bengaluru Varatonal Methods and Structural Optmzaton G. K. Ananthasuresh Professor, Mechancal Engneerng,
More informationChapter 12. Ordinary Differential Equation Boundary Value (BV) Problems
Chapter. Ordnar Dfferental Equaton Boundar Value (BV) Problems In ths chapter we wll learn how to solve ODE boundar value problem. BV ODE s usuall gven wth x beng the ndependent space varable. p( x) q(
More informationProfessor Terje Haukaas University of British Columbia, Vancouver The Q4 Element
Professor Terje Haukaas Unversty of Brtsh Columba, ancouver www.nrsk.ubc.ca The Q Element Ths document consders fnte elements that carry load only n ther plane. These elements are sometmes referred to
More informationAppendix for Causal Interaction in Factorial Experiments: Application to Conjoint Analysis
A Appendx for Causal Interacton n Factoral Experments: Applcaton to Conjont Analyss Mathematcal Appendx: Proofs of Theorems A. Lemmas Below, we descrbe all the lemmas, whch are used to prove the man theorems
More informationThe Feynman path integral
The Feynman path ntegral Aprl 3, 205 Hesenberg and Schrödnger pctures The Schrödnger wave functon places the tme dependence of a physcal system n the state, ψ, t, where the state s a vector n Hlbert space
More informationSolution of Linear System of Equations and Matrix Inversion Gauss Seidel Iteration Method
Soluton of Lnear System of Equatons and Matr Inverson Gauss Sedel Iteraton Method It s another well-known teratve method for solvng a system of lnear equatons of the form a + a22 + + ann = b a2 + a222
More informationNonlinear Overlapping Domain Decomposition Methods
Nonlnear Overlappng Doman Decomposton Methods Xao-Chuan Ca 1 Department of Computer Scence, Unversty of Colorado at Boulder, Boulder, CO 80309, ca@cs.colorado.edu Summary. We dscuss some overlappng doman
More informationPhysics 5153 Classical Mechanics. D Alembert s Principle and The Lagrangian-1
P. Guterrez Physcs 5153 Classcal Mechancs D Alembert s Prncple and The Lagrangan 1 Introducton The prncple of vrtual work provdes a method of solvng problems of statc equlbrum wthout havng to consder the
More informationGeneralized Linear Methods
Generalzed Lnear Methods 1 Introducton In the Ensemble Methods the general dea s that usng a combnaton of several weak learner one could make a better learner. More formally, assume that we have a set
More informationPolynomial Regression Models
LINEAR REGRESSION ANALYSIS MODULE XII Lecture - 6 Polynomal Regresson Models Dr. Shalabh Department of Mathematcs and Statstcs Indan Insttute of Technology Kanpur Test of sgnfcance To test the sgnfcance
More informationBezier curves. Michael S. Floater. August 25, These notes provide an introduction to Bezier curves. i=0
Bezer curves Mchael S. Floater August 25, 211 These notes provde an ntroducton to Bezer curves. 1 Bernsten polynomals Recall that a real polynomal of a real varable x R, wth degree n, s a functon of the
More informationYong Joon Ryang. 1. Introduction Consider the multicommodity transportation problem with convex quadratic cost function. 1 2 (x x0 ) T Q(x x 0 )
Kangweon-Kyungk Math. Jour. 4 1996), No. 1, pp. 7 16 AN ITERATIVE ROW-ACTION METHOD FOR MULTICOMMODITY TRANSPORTATION PROBLEMS Yong Joon Ryang Abstract. The optmzaton problems wth quadratc constrants often
More informationChapter 5. Solution of System of Linear Equations. Module No. 6. Solution of Inconsistent and Ill Conditioned Systems
Numercal Analyss by Dr. Anta Pal Assstant Professor Department of Mathematcs Natonal Insttute of Technology Durgapur Durgapur-713209 emal: anta.bue@gmal.com 1 . Chapter 5 Soluton of System of Lnear Equatons
More informationAsymptotics of the Solution of a Boundary Value. Problem for One-Characteristic Differential. Equation Degenerating into a Parabolic Equation
Nonl. Analyss and Dfferental Equatons, ol., 4, no., 5 - HIKARI Ltd, www.m-har.com http://dx.do.org/.988/nade.4.456 Asymptotcs of the Soluton of a Boundary alue Problem for One-Characterstc Dfferental Equaton
More informationThe Study of Teaching-learning-based Optimization Algorithm
Advanced Scence and Technology Letters Vol. (AST 06), pp.05- http://dx.do.org/0.57/astl.06. The Study of Teachng-learnng-based Optmzaton Algorthm u Sun, Yan fu, Lele Kong, Haolang Q,, Helongang Insttute
More informationLinear Feature Engineering 11
Lnear Feature Engneerng 11 2 Least-Squares 2.1 Smple least-squares Consder the followng dataset. We have a bunch of nputs x and correspondng outputs y. The partcular values n ths dataset are x y 0.23 0.19
More informationConvexity preserving interpolation by splines of arbitrary degree
Computer Scence Journal of Moldova, vol.18, no.1(52), 2010 Convexty preservng nterpolaton by splnes of arbtrary degree Igor Verlan Abstract In the present paper an algorthm of C 2 nterpolaton of dscrete
More informationThe Quadratic Trigonometric Bézier Curve with Single Shape Parameter
J. Basc. Appl. Sc. Res., (3541-546, 01 01, TextRoad Publcaton ISSN 090-4304 Journal of Basc and Appled Scentfc Research www.textroad.com The Quadratc Trgonometrc Bézer Curve wth Sngle Shape Parameter Uzma
More informationNeuro-Adaptive Design - I:
Lecture 36 Neuro-Adaptve Desgn - I: A Robustfyng ool for Dynamc Inverson Desgn Dr. Radhakant Padh Asst. Professor Dept. of Aerospace Engneerng Indan Insttute of Scence - Bangalore Motvaton Perfect system
More informationSupplement: Proofs and Technical Details for The Solution Path of the Generalized Lasso
Supplement: Proofs and Techncal Detals for The Soluton Path of the Generalzed Lasso Ryan J. Tbshran Jonathan Taylor In ths document we gve supplementary detals to the paper The Soluton Path of the Generalzed
More informationLecture Notes on Linear Regression
Lecture Notes on Lnear Regresson Feng L fl@sdueducn Shandong Unversty, Chna Lnear Regresson Problem In regresson problem, we am at predct a contnuous target value gven an nput feature vector We assume
More informationHongyi Miao, College of Science, Nanjing Forestry University, Nanjing ,China. (Received 20 June 2013, accepted 11 March 2014) I)ϕ (k)
ISSN 1749-3889 (prnt), 1749-3897 (onlne) Internatonal Journal of Nonlnear Scence Vol.17(2014) No.2,pp.188-192 Modfed Block Jacob-Davdson Method for Solvng Large Sparse Egenproblems Hongy Mao, College of
More informationLecture 5.8 Flux Vector Splitting
Lecture 5.8 Flux Vector Splttng 1 Flux Vector Splttng The vector E n (5.7.) can be rewrtten as E = AU (5.8.1) (wth A as gven n (5.7.4) or (5.7.6) ) whenever, the equaton of state s of the separable form
More informationErratum: A Generalized Path Integral Control Approach to Reinforcement Learning
Journal of Machne Learnng Research 00-9 Submtted /0; Publshed 7/ Erratum: A Generalzed Path Integral Control Approach to Renforcement Learnng Evangelos ATheodorou Jonas Buchl Stefan Schaal Department of
More informationFACTORIZATION IN KRULL MONOIDS WITH INFINITE CLASS GROUP
C O L L O Q U I U M M A T H E M A T I C U M VOL. 80 1999 NO. 1 FACTORIZATION IN KRULL MONOIDS WITH INFINITE CLASS GROUP BY FLORIAN K A I N R A T H (GRAZ) Abstract. Let H be a Krull monod wth nfnte class
More informationSingular Value Decomposition: Theory and Applications
Sngular Value Decomposton: Theory and Applcatons Danel Khashab Sprng 2015 Last Update: March 2, 2015 1 Introducton A = UDV where columns of U and V are orthonormal and matrx D s dagonal wth postve real
More informationThe Minimum Universal Cost Flow in an Infeasible Flow Network
Journal of Scences, Islamc Republc of Iran 17(2): 175-180 (2006) Unversty of Tehran, ISSN 1016-1104 http://jscencesutacr The Mnmum Unversal Cost Flow n an Infeasble Flow Network H Saleh Fathabad * M Bagheran
More informationHomework Notes Week 7
Homework Notes Week 7 Math 4 Sprng 4 #4 (a Complete the proof n example 5 that s an nner product (the Frobenus nner product on M n n (F In the example propertes (a and (d have already been verfed so we
More informationAppendix B. The Finite Difference Scheme
140 APPENDIXES Appendx B. The Fnte Dfference Scheme In ths appendx we present numercal technques whch are used to approxmate solutons of system 3.1 3.3. A comprehensve treatment of theoretcal and mplementaton
More information2.3 Nilpotent endomorphisms
s a block dagonal matrx, wth A Mat dm U (C) In fact, we can assume that B = B 1 B k, wth B an ordered bass of U, and that A = [f U ] B, where f U : U U s the restrcton of f to U 40 23 Nlpotent endomorphsms
More informationNumerical Simulation of One-Dimensional Wave Equation by Non-Polynomial Quintic Spline
IOSR Journal of Matematcs (IOSR-JM) e-issn: 78-578, p-issn: 319-765X. Volume 14, Issue 6 Ver. I (Nov - Dec 018), PP 6-30 www.osrournals.org Numercal Smulaton of One-Dmensonal Wave Equaton by Non-Polynomal
More informationCOMPUTATIONAL METHODS AND ALGORITHMS Vol. II - Finite Element Method - Jacques-Hervé SAIAC
COMPUTATIONAL METHODS AND ALGORITHMS Vol. II - Fnte Element Method - Jacques-Hervé SAIAC FINITE ELEMENT METHOD Jacques-Hervé SAIAC Départment de Mathématques, Conservatore Natonal des Arts et Méters, Pars,
More informationA PROCEDURE FOR SIMULATING THE NONLINEAR CONDUCTION HEAT TRANSFER IN A BODY WITH TEMPERATURE DEPENDENT THERMAL CONDUCTIVITY.
Proceedngs of the th Brazlan Congress of Thermal Scences and Engneerng -- ENCIT 006 Braz. Soc. of Mechancal Scences and Engneerng -- ABCM, Curtba, Brazl,- Dec. 5-8, 006 A PROCEDURE FOR SIMULATING THE NONLINEAR
More informationAn efficient algorithm for multivariate Maclaurin Newton transformation
Annales UMCS Informatca AI VIII, 2 2008) 5 14 DOI: 10.2478/v10065-008-0020-6 An effcent algorthm for multvarate Maclaurn Newton transformaton Joanna Kapusta Insttute of Mathematcs and Computer Scence,
More informationSolutions to exam in SF1811 Optimization, Jan 14, 2015
Solutons to exam n SF8 Optmzaton, Jan 4, 25 3 3 O------O -4 \ / \ / The network: \/ where all lnks go from left to rght. /\ / \ / \ 6 O------O -5 2 4.(a) Let x = ( x 3, x 4, x 23, x 24 ) T, where the varable
More informationVector Norms. Chapter 7 Iterative Techniques in Matrix Algebra. Cauchy-Bunyakovsky-Schwarz Inequality for Sums. Distances. Convergence.
Vector Norms Chapter 7 Iteratve Technques n Matrx Algebra Per-Olof Persson persson@berkeley.edu Department of Mathematcs Unversty of Calforna, Berkeley Math 128B Numercal Analyss Defnton A vector norm
More informationLagrangian Field Theory
Lagrangan Feld Theory Adam Lott PHY 391 Aprl 6, 017 1 Introducton Ths paper s a summary of Chapter of Mandl and Shaw s Quantum Feld Theory [1]. The frst thng to do s to fx the notaton. For the most part,
More informationMATH 241B FUNCTIONAL ANALYSIS - NOTES EXAMPLES OF C ALGEBRAS
MATH 241B FUNCTIONAL ANALYSIS - NOTES EXAMPLES OF C ALGEBRAS These are nformal notes whch cover some of the materal whch s not n the course book. The man purpose s to gve a number of nontrval examples
More informationModule 3 LOSSY IMAGE COMPRESSION SYSTEMS. Version 2 ECE IIT, Kharagpur
Module 3 LOSSY IMAGE COMPRESSION SYSTEMS Verson ECE IIT, Kharagpur Lesson 6 Theory of Quantzaton Verson ECE IIT, Kharagpur Instructonal Objectves At the end of ths lesson, the students should be able to:
More informationTensor Analysis. For orthogonal curvilinear coordinates, ˆ ˆ (98) Expanding the derivative, we have, ˆ. h q. . h q h q
For orthogonal curvlnear coordnates, eˆ grad a a= ( aˆ ˆ e). h q (98) Expandng the dervatve, we have, eˆ aˆ ˆ e a= ˆ ˆ a h e + q q 1 aˆ ˆ ˆ a e = ee ˆˆ ˆ + e. h q h q Now expandng eˆ / q (some of the detals
More information8.4 COMPLEX VECTOR SPACES AND INNER PRODUCTS
SECTION 8.4 COMPLEX VECTOR SPACES AND INNER PRODUCTS 493 8.4 COMPLEX VECTOR SPACES AND INNER PRODUCTS All the vector spaces you have studed thus far n the text are real vector spaces because the scalars
More information