Topology Optimization of Elastic Material Microstructures with Classic Models of Micromechanics
|
|
- Gary Holland
- 6 years ago
- Views:
Transcription
1 COMPUTATIONAL MECHANICS WCCM VI n conjuncton wth APCOM 4, Sept. 5-1, 24, Bejng, Chna 24 Tsnghua Unversty Press & Sprnger-Verlag Topology Optmzaton of Elastc Materal Mcrostructures wth Classc Models of Mcromechancs W.H. Zhang 1*, S.P. Sun 1, L. Wang 1, D. Wang 2 1 Insttute of Mechatronc Engneerng, Northwestern Polytechncal Unversty, X an, 7172, Shaanx, Chna 2 Insttute of Aeronautcs, Northwestern Polytechncal Unversty, X an, 7172, Shaanx,Chna e-mal: zhangwh@nwpu.edu.cn, shpngsun@163.com, lewang9@hotmal.com, wangdng6682@sna.com Abstract Nowadays, topology optmzaton consttutes a basc and powerful method for the desgn of materal mcrostructures. Recent advances show that ths method s manly based on the homogenzaton method that deals wth perodc unt cells usng the asymptotc expanson. Ths formulaton requres that the homogenzaton method be mplemented as an addtonal module n the context of a fnte element system to predct the effectve propertes of the mcrostructure. Theoretcally, the homogenzaton method s known to be vald only when the characterstc scale-sze of the concerned unt cell s small enough. In ths paper, topology optmzaton of the materal mcrostructure s carred out based on the classc model of mcromechancs wth Drchlet boundary condtons. The stffness of the mcrostructure s maxmzed wth the gven amount of materals. It s shown that ths approach s effcent to gve rse to satsfactory materal layouts n dfferent loadng condtons from the engneerng vewpont. Numercally, the advantage of ths approach les n that topology optmzaton of the mcrostructure can be performed n the same way as ts orgnal verson used for macrostructures wthout needs of addtonal software programmng. The key s to retan sutable boundary condtons n the fnte element modelng of the unt cell. Fnally, numercal tests are used to llustrate the valdty of the proposed method. Key words: topology optmzaton, materal desgn, mcrostructure, homogenzaton method INTRODUCTION Topology optmzaton s recognzed as an effcent tool n materals layout desgn. The mportance s based on the fact that the topology, shape, scale and volume fractons of the consttuents for a mcrostructure have a determnant nfluence upon ts effectve propertes. By topology optmzaton, structures can be, on one hand, effectvely optmzed n stffness, stablty and dynamc responses wth the gven amount of materals n the macro-scale; and on the other hand, materals can be optmally talored ether to attan prescrbed even extreme propertes or to provde nnovatve mcrostructures n the mcroscopc scale. For example, desgns of mcrostructures wth negatve Posson rato, zero thermal expanson coeffcents [1]. Topology optmzaton becomes thus a common technque to carry out smultaneous optmzaton of materals and structures. Hstorcally, the classcal model of mcromechancs (CMM) and the homogenzaton method (HM) consttute two basc approaches. The former s based on Drchlet or Neumann boundary condtons [2] whereas the latter s based on the asymptotc expanson wth perodc condtons of the unt cell [3]. Actually, desgnng mcrostructure s often carred out by ntegratng topology optmzaton wth the homogenzaton method. To ths end, a fnte element mplementaton has to be made for both evaluatons and senstvty analyss of macroscopc propertes. Conversely, the classcal model of mcromechancs s rather smple for the numercal predcton of effectve propertes of mcrostructures. But t seems to be not well exploted and combned wth topology optmzaton. 1 1
2 In ths paper, the classcal model of mcromechancs s used together wth topology optmzaton procedure to desgn the mcrostructure of materals. The procedure can be smply carred out as the same as for macrostructures wthout further mplementaton work. Moreover, after the nvolved mcrostructure s optmzed, attempts are made to perform topology optmzaton of macrostructures wth orthotropc materal propertes. CLASSICAL MODEL OF MICROMECHANICS WITH DIRICHLET BOUNDARY CONDITIONS v=1 u=1/2 u=1 v=1/2 (2) (3) Fg. 1 Three loadng cases wth unt deformaton boundary condtons Consder a 2-D unt cell shown n Fg. 1. The equvalent stffness matrx conssts of E1111 E1122 E = E 2211 E E The equvalent macroscopc stffness matrx can be numercally predcted by usng the fnte element method. Under baxal and shear solctatons of ntal unt deformatons along the boundary, the deformaton energy and reacton forces along the boundary of the unt cell can be easly evaluated. Consequently, one can wrte (2) E 1111 = C, 2222 C (3) E =, E 1212 = C, 1122 = E 2211 = σ 22 E (2) Note that symbol C and the superscrpt corresponds to the deformaton energy and the load case, respectvely. Alternatvely, f the homogenzaton method s used, the equvalent macroscopc stffness matrx wll be evaluated as the ntegral over the unt cell E jkl 1 χ p = Y ( Ejkl Ejpq ) dy Y y kl q (3) Where Y and E jkl denote the volume of the unt cell and the rgdty coeffcents of the nvolved consttuents, respectvely. In ths formulaton, the unt cell has to be specfed wth perodc dsplacement boundary condtons and the loadng measurng the spatal varaton of materal propertes over the unt cell has to be defned when 2 2
3 evaluatng the ntermedate dsplacement feld kl χ p by the fnte element method. For 2D and 3D unt cells, evaluatons of kl χ p wll be carred out under 3 and 6 loadng cases, respectvely wth kl = 11,22,12 or kl = 11,22,33,12,13, 23. From (3), t s clearly seen that each materal consttuent n the unt cell has ts proper contrbuton to the equvalent propertes. In fact, the frst part of the ntegral n (3) corresponds to a volume fracton average whereas the second one can be regarded as a correcton term dependng upon the shape, confguraton and poston of consttuents nsde the unt cell. TOPOLOGY OPTIMIZATION PROCEDURE OF THE MICROSTRUCTURE As llustrated below, both above methods agree well n precson for the numercal predcton of the stffness matrx. Snce prescrbed dsplacement boundary condtons are appled, the stffest mcrostructure,.e, the maxmzaton of dagonal terms of the stffness matrx s the same as the maxmzaton of deformaton energy n 3 loadng cases. Mathematcally, when the unt cell s meshed wth fnte elements whose pseudo denstes X are used as desgn varables, the optmzaton problem s expressed as Max C ( k ) n ( X ) V ( X ) = xv V = 1 TVa ( X ) P < δ x 1 = 1, n (4) where V(X) s the total materal volume lmted by ts upper bound V over the unt cell. A small value of δ =1-5 s used to avod the sngularty of the elementary stffness matrx durng optmzaton. TV a denotes the total varaton (TV) control whose role s, lke the permeter control, to regularze the pattern of the materal layout. The role of upper bound P s twofold. On one hand, the avodance of checkerboards can be ensured wth small values of P, on the other hand, ts relaxaton wll elmnate ntermedate values between and 1 for densty varables. Here, we adopt TV 2 whose expresson s M 2 TV2 = lk ( x x j ) (5) k = 1 lk denotes the edge length of the kth nterface between two adjacent fnte elements and j. Jumps of materal densty varatons at all adjacent elements are controlled by the upper bound P. Detals about ths control can be found n [4]. To solve problem (4), the deformaton energy of the unt cell assocated wth prescrbed dsplacements shown n Fg. 1 wll be frstly evaluated at current values of densty varables. Thereafter, senstvty analyss of the deformaton energy s evaluated n such a way that prescrbed dsplacements are regarded as equvalent prescrbed loads. From ths vewpont, the maxmzaton problem (4) becomes a mnmzaton one and the senstvty of the deformaton energy wll be a scalng of the concerned element deformaton energy under equvalent prescrbed loads. Namely, C x ( k ) = p x C ( k ) (6) where p s the exponent used as the penalty of the power law for the element stffness matrx (see Ref. [4]). NUMERICAL TESTS In ths secton, nvestgatons wll be focused on the followng aspects: 3 3
4 1) Comparson between the homogenzaton method and the classc model of mcromechancs. 2) Topology optmzaton of mcrostructures 3) Numercal computng of macrostructure flexblty wth and wthout homogenzaton 4) Topology optmzaton of macrostructures consstng of mcrostructures wth orthotropc propertes 1. Numercal predctons of the stffness matrx of the mcrostructure A set of unt cells studed prevously n [5] wth dfferent scale szes and elastc constants of consttuents s gven n Fg. 2. Both CMM and HM methods are appled, respectvely to evaluate the macroscopc stffness matrx. Note that snce boundary condtons used here are dfferent from those n [5], our results wll be dfferent from those obtaned n [5] for the stffness matrx. Fg. 2 Illustraton of mcrostructures of unt cells Assume that the mcrostructure s made of two sotropc materals ndcated by red and blue colors, respectvely. The volume fracton s 5%. Case 1: materal propertes are E1 = 1, E2 = 1; ν 1 =ν 2 =.3 Stffness matrces obtaned wth the classc model (2) and the homogenzaton method (3) are as follows, respectvely (7) (CMM) (HM) Case 2: materal propertes are E 1 = 1, E 2 = 1 ; ν 1 =ν 2 =.3 Stffness matrces obtaned wth the classc model (2) and the homogenzaton method (3) are as follows, respectvely (CMM) (HM) From above results, t turns out that the scale-sze of the unt cell has no nfluence upon the results wth the current mcromechancs model and that both methods produce almost the same stffness matrx. Therefore, both methods can be used n the numercal predcton of the equvalent stffness matrx. 2.Topology optmzaton of mcrostructures Based on three boundary condtons gven n Fg. 1, the mcrostructure wll be desgned wth a two-phase of vod and sotropc materal. To do ths, relatons n (2) wll be used to evaluate stffness terms, optmzaton wll be carred out by means of (4) and senstvty analyss wll be performed by means of (6). In these tests, volume fractons of 3% and 5% wll be taken nto account for the sold materal n the unt cell, respectvely. As plotted n Fg. 3, results of materal layouts are gven n columns 1 and 3. To have a global vew about the dstrbuton of mcrostructures, columns 2 and 4 provde a perodc repetton of the related unt cell. From the engneerng vewpont, the above mcrostructures possess ndeed the maxmum stffness n tracton and shear. (8) 4 4
5 Volume fracton 3% Volume fracton 5% Fg. 3 Topology optmzaton of mcrostructures Now, consder a combned load case of b-objectve optmzaton wth horzontal tracton ( u =. 1 ) and shear boundary condtons ( u = v =.1). Gven a volume fracton of 5%, the desgn soluton of the mcrostructure s shown n Fg. 4. One can see that the TV control s effectve to regularze the desgn pattern. Fg. 4 Topology optmzaton of mcrostructures In ths case, equvalent stffness matrces obtaned are as follows ( 9 ) 3.Flexblty analyss and topology optmzaton of the macrostructure As s known, topology optmzaton wth stffness desgn n the macroscopc level depends strongly upon the accuracy of flexblty analyss. To nvestgate the nfluence of the mcrostructure, consder now a cantlevered beam wth perforated holes dstrbuted perodcally over the doman. As shown n Fg. 5, the beam s loaded by a 5 5
6 vertcal force on the mddle pont of the rght sde. Suppose that Young s modulus of the beam materal s E 1 = Perforaton 4 8, E 1 = (2) E 1 /E 2 = /1.e-3 (3) Perforaton 8 16, E 1 = (4) Homogenzaton Fg. 5 Flexblty analyss In Fg. 5, plot corresponds to the dsplacement result when the structure has a perforaton of 4 8 and s drectly dscretzed and evaluated by the fnte element method. Plot (2) corresponds to the dsplacement result for whch vods are all flled wth softenng materals (E 2 =1.e-3). Plot (3) corresponds to the result wth a perforaton of Plot (4) corresponds to the result for whch an equvalent sold sheet of orthotropc materal propertes s used wth the followng homogenzed stffness matrx It s seen that wth an ncrease of perforatons, the dsplacement response has the tendency of convergng to the homogenzed result. Now, consder the topology optmzaton of mnmzng the complance of the cantlevered beam n the macroscopc level. By means of the above equvalent stffness matrx, the power law wth exponent p=4 s used to penalze the above stffness matrx n the fnte element formulaton. A macroscopc volume fracton of 6% s used. The desgn soluton s shown n Fg. 5. Fg.5 Topology optmzaton of the macrostructure 6 6
7 CONCLUSIONS In ths paper, the classc model of mcromechancs and homogenzaton method are nvestgated. It s shown that both methods have a good coherence for the numercal predcton of effectve propertes of the mcrostructure f boundary condtons are sutably used. Ths coherence provdes an alternatve approach of ntegratng the classcal model wth the topology optmzaton methodology. Under dsplacement boundary condtons, deformaton energes assocated wth dagonal terms of the equvalent stffness matrx can be maxmzed to fnd the optmal layout of the stffest mcrostructures. Wth the avalable materals, topology optmzaton of mcrostructures can be performed n the same way as for structures of macro-scale. Furthermore, the last example provdes an dea of jonng mcrostructure optmzaton wth macrostructure optmzaton. It s shown that topology optmzaton of the macrostructure can be carred out based on the desgn result of mcrostructures. Ths concept wll promote the methodology of smultaneous desgn of materals and structures. Acknowledgements The support of the Natonal Natural Scence Foundaton of Chna (Grant No ) s gratefully acknowledged. REFERENCES [1] O. Sgmund, S. Torquato, Desgn of materals wth extreme thermal expanson usng a three-phase to[ology optmzatuon method, J. Mech. Phys. Solds., 45, (1997), [2] Z. Hashn, Analyss of composte materals, ASME Journal of Appled Mechancs, 5, (1983), [3] A. Bensoussan, J.L. Lons, G. Papancolaou, Asymptotc analyss for perodc structures, North Holland, Amsterdam (1978). [4] W.H. Zhang, P. Duysnx, Dual approach usng a varant permeter constrant and effcent sub-teraton scheme for topology optmzaton, Computers & Structures, 81(22/23), [5] S. Pecullan. L.V. Gbansky, S. Torquato, Scale effects on the elastc behavor of perodc and herarchcal two-dmensonal compostes, J. Mech. Phys. Solds., 47, (1999),
OPTIMAL DESIGN OF VISCOELASTIC COMPOSITES WITH PERIODIC MICROSTRUCTURES
OPTIMAL DESIN OF VISCOELASTIC COMPOSITES WITH PERIODIC MICROSTRUCTURES eong-moo 1, Sang-Hoon Park 2, and Sung-Ke oun 2 1 Space Technology Dvson, Korea Aerospace Research Insttute, usung P.O. Box 3, Taejon,
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 informationSecond Order Analysis
Second Order Analyss In the prevous classes we looked at a method that determnes the load correspondng to a state of bfurcaton equlbrum of a perfect frame by egenvalye analyss The system was assumed to
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 informationNUMERICAL EVALUATION OF PERIODIC BOUNDARY CONDITION ON THERMO-MECHANICAL PROBLEM USING HOMOGENIZATION METHOD
THE 9 TH INTERNATIONAL CONFERENCE ON COMPOSITE MATERIALS NUMERICAL EVALUATION OF PERIODIC BOUNDAR CONDITION ON THERMO-MECHANICAL PROBLEM USING HOMOGENIZATION METHOD M.R.E. Nasuton *, N. Watanabe, A. Kondo,2
More informationCHAPTER 9 CONCLUSIONS
78 CHAPTER 9 CONCLUSIONS uctlty and structural ntegrty are essentally requred for structures subjected to suddenly appled dynamc loads such as shock loads. Renforced Concrete (RC), the most wdely used
More informationDUE: WEDS FEB 21ST 2018
HOMEWORK # 1: FINITE DIFFERENCES IN ONE DIMENSION DUE: WEDS FEB 21ST 2018 1. Theory Beam bendng s a classcal engneerng analyss. The tradtonal soluton technque makes smplfyng assumptons such as a constant
More informationA comprehensive study: Boundary conditions for representative volume elements (RVE) of composites
Insttute of Structural Mechancs A comprehensve study: Boundary condtons for representatve volume elements (RVE) of compostes Srhar Kurukur A techncal report on homogenzaton technques A comprehensve study:
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 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 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 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 informationA Local Variational Problem of Second Order for a Class of Optimal Control Problems with Nonsmooth Objective Function
A Local Varatonal Problem of Second Order for a Class of Optmal Control Problems wth Nonsmooth Objectve Functon Alexander P. Afanasev Insttute for Informaton Transmsson Problems, Russan Academy of Scences,
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 information829. An adaptive method for inertia force identification in cantilever under moving mass
89. An adaptve method for nerta force dentfcaton n cantlever under movng mass Qang Chen 1, Mnzhuo Wang, Hao Yan 3, Haonan Ye 4, Guola Yang 5 1,, 3, 4 Department of Control and System Engneerng, Nanng Unversty,
More informationAdjoint Methods of Sensitivity Analysis for Lyapunov Equation. Boping Wang 1, Kun Yan 2. University of Technology, Dalian , P. R.
th World Congress on Structural and Multdscplnary Optmsaton 7 th - th, June 5, Sydney Australa Adjont Methods of Senstvty Analyss for Lyapunov Equaton Bopng Wang, Kun Yan Department of Mechancal and Aerospace
More informationSimulation of 2D Elastic Bodies with Randomly Distributed Circular Inclusions Using the BEM
Smulaton of 2D Elastc Bodes wth Randomly Dstrbuted Crcular Inclusons Usng the BEM Zhenhan Yao, Fanzhong Kong 2, Xaopng Zheng Department of Engneerng Mechancs 2 State Key Lab of Automotve Safety and Energy
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 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 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 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 informationTHE EFFECT OF TORSIONAL RIGIDITY BETWEEN ELEMENTS ON FREE VIBRATIONS OF A TELESCOPIC HYDRAULIC CYLINDER SUBJECTED TO EULER S LOAD
Journal of Appled Mathematcs and Computatonal Mechancs 7, 6(3), 7- www.amcm.pcz.pl p-issn 99-9965 DOI:.75/jamcm.7.3. e-issn 353-588 THE EFFECT OF TORSIONAL RIGIDITY BETWEEN ELEMENTS ON FREE VIBRATIONS
More informationIndeterminate pin-jointed frames (trusses)
Indetermnate pn-jonted frames (trusses) Calculaton of member forces usng force method I. Statcal determnacy. The degree of freedom of any truss can be derved as: w= k d a =, where k s the number of all
More informationOne-sided finite-difference approximations suitable for use with Richardson extrapolation
Journal of Computatonal Physcs 219 (2006) 13 20 Short note One-sded fnte-dfference approxmatons sutable for use wth Rchardson extrapolaton Kumar Rahul, S.N. Bhattacharyya * Department of Mechancal Engneerng,
More informationχ x B E (c) Figure 2.1.1: (a) a material particle in a body, (b) a place in space, (c) a configuration of the body
Secton.. Moton.. The Materal Body and Moton hyscal materals n the real world are modeled usng an abstract mathematcal entty called a body. Ths body conssts of an nfnte number of materal partcles. Shown
More informationOFF-AXIS MECHANICAL PROPERTIES OF FRP COMPOSITES
ICAMS 204 5 th Internatonal Conference on Advanced Materals and Systems OFF-AXIS MECHANICAL PROPERTIES OF FRP COMPOSITES VLAD LUPĂŞTEANU, NICOLAE ŢĂRANU, RALUCA HOHAN, PAUL CIOBANU Gh. Asach Techncal Unversty
More informationGeneral displacement arch-cantilever element method for stress analysis of arch dam
Water Scence and Engneerng, 009, (): 3-4 do:0.388/j.ssn.674-370.009.0.004 http://kkb.hhu.edu.cn e-mal: wse@hhu.edu.cn General dsplacement arch-cantlever element method for stress analyss of arch dam Hao
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 informationGlobal Sensitivity. Tuesday 20 th February, 2018
Global Senstvty Tuesday 2 th February, 28 ) Local Senstvty Most senstvty analyses [] are based on local estmates of senstvty, typcally by expandng the response n a Taylor seres about some specfc values
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 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 informationComputer Based Porosity Design by Multi Phase Topology Optimization
Computer Based Porosty Desgn by Mult Phase Topology Optmzaton Andreas Burbles and Matthas Busse Fraunhofer-Insttut für Fertgungstechnk und Angewandte Materalforschung - IFAM Wener Str. 12, 28359 Bremen,
More informationDr. Shalabh Department of Mathematics and Statistics Indian Institute of Technology Kanpur
Analyss of Varance and Desgn of Exerments-I MODULE III LECTURE - 2 EXPERIMENTAL DESIGN MODELS Dr. Shalabh Deartment of Mathematcs and Statstcs Indan Insttute of Technology Kanur 2 We consder the models
More informationSTATIC ANALYSIS OF TWO-LAYERED PIEZOELECTRIC BEAMS WITH IMPERFECT SHEAR CONNECTION
STATIC ANALYSIS OF TWO-LERED PIEZOELECTRIC BEAMS WITH IMPERFECT SHEAR CONNECTION Ákos József Lengyel István Ecsed Assstant Lecturer Emertus Professor Insttute of Appled Mechancs Unversty of Mskolc Mskolc-Egyetemváros
More informationAn Interactive Optimisation Tool for Allocation Problems
An Interactve Optmsaton ool for Allocaton Problems Fredr Bonäs, Joam Westerlund and apo Westerlund Process Desgn Laboratory, Faculty of echnology, Åbo Aadem Unversty, uru 20500, Fnland hs paper presents
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 informationNON-CENTRAL 7-POINT FORMULA IN THE METHOD OF LINES FOR PARABOLIC AND BURGERS' EQUATIONS
IJRRAS 8 (3 September 011 www.arpapress.com/volumes/vol8issue3/ijrras_8_3_08.pdf NON-CENTRAL 7-POINT FORMULA IN THE METHOD OF LINES FOR PARABOLIC AND BURGERS' EQUATIONS H.O. Bakodah Dept. of Mathematc
More informationLinear Approximation with Regularization and Moving Least Squares
Lnear Approxmaton wth Regularzaton and Movng Least Squares Igor Grešovn May 007 Revson 4.6 (Revson : March 004). 5 4 3 0.5 3 3.5 4 Contents: Lnear Fttng...4. Weghted Least Squares n Functon Approxmaton...
More informationTHE SMOOTH INDENTATION OF A CYLINDRICAL INDENTOR AND ANGLE-PLY LAMINATES
THE SMOOTH INDENTATION OF A CYLINDRICAL INDENTOR AND ANGLE-PLY LAMINATES W. C. Lao Department of Cvl Engneerng, Feng Cha Unverst 00 Wen Hwa Rd, Tachung, Tawan SUMMARY: The ndentaton etween clndrcal ndentor
More informationProblem Set 9 Solutions
Desgn and Analyss of Algorthms May 4, 2015 Massachusetts Insttute of Technology 6.046J/18.410J Profs. Erk Demane, Srn Devadas, and Nancy Lynch Problem Set 9 Solutons Problem Set 9 Solutons Ths problem
More informationModule 1 : The equation of continuity. Lecture 1: Equation of Continuity
1 Module 1 : The equaton of contnuty Lecture 1: Equaton of Contnuty 2 Advanced Heat and Mass Transfer: Modules 1. THE EQUATION OF CONTINUITY : Lectures 1-6 () () () (v) (v) Overall Mass Balance Momentum
More informationSpeeding up Computation of Scalar Multiplication in Elliptic Curve Cryptosystem
H.K. Pathak et. al. / (IJCSE) Internatonal Journal on Computer Scence and Engneerng Speedng up Computaton of Scalar Multplcaton n Ellptc Curve Cryptosystem H. K. Pathak Manju Sangh S.o.S n Computer scence
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 informationOne Dimensional Axial Deformations
One Dmensonal al Deformatons In ths secton, a specfc smple geometr s consdered, that of a long and thn straght component loaded n such a wa that t deforms n the aal drecton onl. The -as s taken as the
More informationDESIGN OPTIMIZATION OF CFRP RECTANGULAR BOX SUBJECTED TO ARBITRARY LOADINGS
Munch, Germany, 26-30 th June 2016 1 DESIGN OPTIMIZATION OF CFRP RECTANGULAR BOX SUBJECTED TO ARBITRARY LOADINGS Q.T. Guo 1*, Z.Y. L 1, T. Ohor 1 and J. Takahash 1 1 Department of Systems Innovaton, School
More informationMACHINE APPLIED MACHINE LEARNING LEARNING. Gaussian Mixture Regression
11 MACHINE APPLIED MACHINE LEARNING LEARNING MACHINE LEARNING Gaussan Mture Regresson 22 MACHINE APPLIED MACHINE LEARNING LEARNING Bref summary of last week s lecture 33 MACHINE APPLIED MACHINE LEARNING
More informationModule 9. Lecture 6. Duality in Assignment Problems
Module 9 1 Lecture 6 Dualty n Assgnment Problems In ths lecture we attempt to answer few other mportant questons posed n earler lecture for (AP) and see how some of them can be explaned through the concept
More informationFUZZY GOAL PROGRAMMING VS ORDINARY FUZZY PROGRAMMING APPROACH FOR MULTI OBJECTIVE PROGRAMMING PROBLEM
Internatonal Conference on Ceramcs, Bkaner, Inda Internatonal Journal of Modern Physcs: Conference Seres Vol. 22 (2013) 757 761 World Scentfc Publshng Company DOI: 10.1142/S2010194513010982 FUZZY GOAL
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 informationInductance Calculation for Conductors of Arbitrary Shape
CRYO/02/028 Aprl 5, 2002 Inductance Calculaton for Conductors of Arbtrary Shape L. Bottura Dstrbuton: Internal Summary In ths note we descrbe a method for the numercal calculaton of nductances among conductors
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 informationWinter 2008 CS567 Stochastic Linear/Integer Programming Guest Lecturer: Xu, Huan
Wnter 2008 CS567 Stochastc Lnear/Integer Programmng Guest Lecturer: Xu, Huan Class 2: More Modelng Examples 1 Capacty Expanson Capacty expanson models optmal choces of the tmng and levels of nvestments
More informationBuckling analysis of single-layered FG nanoplates on elastic substrate with uneven porosities and various boundary conditions
IOSR Journal of Mechancal and Cvl Engneerng (IOSR-JMCE) e-issn: 78-1684,p-ISSN: 30-334X, Volume 15, Issue 5 Ver. IV (Sep. - Oct. 018), PP 41-46 www.osrjournals.org Bucklng analyss of sngle-layered FG nanoplates
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 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 informationHomogenization of reaction-diffusion processes in a two-component porous medium with a non-linear flux-condition on the interface
Homogenzaton of reacton-dffuson processes n a two-component porous medum wth a non-lnear flux-condton on the nterface Internatonal Conference on Numercal and Mathematcal Modelng of Flow and Transport n
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 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 informationOptimal Control of Temperature in Fluid Flow
Kawahara Lab. 5 March. 27 Optmal Control of Temperature n Flud Flow Dasuke YAMAZAKI Department of Cvl Engneerng, Chuo Unversty Kasuga -3-27, Bunkyou-ku, Tokyo 2-855, Japan E-mal : d33422@educ.kc.chuo-u.ac.jp
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 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 informationLecture 7: Boltzmann distribution & Thermodynamics of mixing
Prof. Tbbtt Lecture 7 etworks & Gels Lecture 7: Boltzmann dstrbuton & Thermodynamcs of mxng 1 Suggested readng Prof. Mark W. Tbbtt ETH Zürch 13 März 018 Molecular Drvng Forces Dll and Bromberg: Chapters
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 informationLecture 8 Modal Analysis
Lecture 8 Modal Analyss 16.0 Release Introducton to ANSYS Mechancal 1 2015 ANSYS, Inc. February 27, 2015 Chapter Overvew In ths chapter free vbraton as well as pre-stressed vbraton analyses n Mechancal
More informationSolution Thermodynamics
Soluton hermodynamcs usng Wagner Notaton by Stanley. Howard Department of aterals and etallurgcal Engneerng South Dakota School of nes and echnology Rapd Cty, SD 57701 January 7, 001 Soluton hermodynamcs
More informationConstitutive Modelling of Superplastic AA-5083
TECHNISCHE MECHANIK, 3, -5, (01, 1-6 submtted: September 19, 011 Consttutve Modellng of Superplastc AA-5083 G. Gulano In ths study a fast procedure for determnng the constants of superplastc 5083 Al alloy
More informationTHEOREMS OF QUANTUM MECHANICS
THEOREMS OF QUANTUM MECHANICS In order to develop methods to treat many-electron systems (atoms & molecules), many of the theorems of quantum mechancs are useful. Useful Notaton The matrx element A mn
More informationMAE140 - Linear Circuits - Winter 16 Final, March 16, 2016
ME140 - Lnear rcuts - Wnter 16 Fnal, March 16, 2016 Instructons () The exam s open book. You may use your class notes and textbook. You may use a hand calculator wth no communcaton capabltes. () You have
More informationLecture 14: Forces and Stresses
The Nuts and Bolts of Frst-Prncples Smulaton Lecture 14: Forces and Stresses Durham, 6th-13th December 2001 CASTEP Developers Group wth support from the ESF ψ k Network Overvew of Lecture Why bother? Theoretcal
More informationComposite Hypotheses testing
Composte ypotheses testng In many hypothess testng problems there are many possble dstrbutons that can occur under each of the hypotheses. The output of the source s a set of parameters (ponts n a parameter
More informationTHE IDENTIFICATION OF MATERIAL PARAMETERS IN NONLINEAR DEFORMATION MODELS OF METALLIC-PLASTIC CYLINDRICAL SHELLS UNDER PULSED LOADING
Materals Physcs and Mechancs (2015) 66-70 Receved: March 27, 2015 THE IDENTIFICATION OF MATERIAL PARAMETERS IN NONLINEAR DEFORMATION MODELS OF METALLIC-PLASTIC CYLINDRICAL SHELLS UNDER PULSED LOADING N.А.
More informationGeorgia Tech PHYS 6124 Mathematical Methods of Physics I
Georga Tech PHYS 624 Mathematcal Methods of Physcs I Instructor: Predrag Cvtanovć Fall semester 202 Homework Set #7 due October 30 202 == show all your work for maxmum credt == put labels ttle legends
More informationElectrical double layer: revisit based on boundary conditions
Electrcal double layer: revst based on boundary condtons Jong U. Km Department of Electrcal and Computer Engneerng, Texas A&M Unversty College Staton, TX 77843-318, USA Abstract The electrcal double layer
More informationStructure and Drive Paul A. Jensen Copyright July 20, 2003
Structure and Drve Paul A. Jensen Copyrght July 20, 2003 A system s made up of several operatons wth flow passng between them. The structure of the system descrbes the flow paths from nputs to outputs.
More informationAn identification algorithm of model kinetic parameters of the interfacial layer growth in fiber composites
IOP Conference Seres: Materals Scence and Engneerng PAPER OPE ACCESS An dentfcaton algorthm of model knetc parameters of the nterfacal layer growth n fber compostes o cte ths artcle: V Zubov et al 216
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 informationA Fast Computer Aided Design Method for Filters
2017 Asa-Pacfc Engneerng and Technology Conference (APETC 2017) ISBN: 978-1-60595-443-1 A Fast Computer Aded Desgn Method for Flters Gang L ABSTRACT *Ths paper presents a fast computer aded desgn method
More informationIrregular vibrations in multi-mass discrete-continuous systems torsionally deformed
(2) 4 48 Irregular vbratons n mult-mass dscrete-contnuous systems torsonally deformed Abstract In the paper rregular vbratons of dscrete-contnuous systems consstng of an arbtrary number rgd bodes connected
More informationCHAPTER 14 GENERAL PERTURBATION THEORY
CHAPTER 4 GENERAL PERTURBATION THEORY 4 Introducton A partcle n orbt around a pont mass or a sphercally symmetrc mass dstrbuton s movng n a gravtatonal potental of the form GM / r In ths potental t moves
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 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 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 information(Online First)A Lattice Boltzmann Scheme for Diffusion Equation in Spherical Coordinate
Internatonal Journal of Mathematcs and Systems Scence (018) Volume 1 do:10.494/jmss.v1.815 (Onlne Frst)A Lattce Boltzmann Scheme for Dffuson Equaton n Sphercal Coordnate Debabrata Datta 1 *, T K Pal 1
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 informationMathematical Modeling and Numerical Simulation of Smart Structures Controlled by Piezoelectric Wafers and Fibers
Mathematcal Modelng and Numercal Smulaton of Smart Structures Controlled by Pezoelectrc Wafers and Fbers U. Gabbert #, W. Kreher*, H. Köppe # # Insttut für Mechank, Otto-von-Guercke Unverstät Magdeburg
More informationEvolutionary Algorithm in Identification of Stochastic Parameters of Laminates
Evolutonary Algorthm n Identfcaton of Stochastc Parameters of Lamnates Potr Orantek 1, Wtold Beluch 1 and Tadeusz Burczyńsk 1,2 1 Department for Strength of Materals and Computatonal Mechancs, Slesan Unversty
More informationDETERMINATION OF TEMPERATURE DISTRIBUTION FOR ANNULAR FINS WITH TEMPERATURE DEPENDENT THERMAL CONDUCTIVITY BY HPM
Ganj, Z. Z., et al.: Determnaton of Temperature Dstrbuton for S111 DETERMINATION OF TEMPERATURE DISTRIBUTION FOR ANNULAR FINS WITH TEMPERATURE DEPENDENT THERMAL CONDUCTIVITY BY HPM by Davood Domr GANJI
More informationCOS 521: Advanced Algorithms Game Theory and Linear Programming
COS 521: Advanced Algorthms Game Theory and Lnear Programmng Moses Charkar February 27, 2013 In these notes, we ntroduce some basc concepts n game theory and lnear programmng (LP). We show a connecton
More informationGEO-SLOPE International Ltd, Calgary, Alberta, Canada Vibrating Beam
GEO-SLOPE Internatonal Ltd, Calgary, Alberta, Canada www.geo-slope.com Introducton Vbratng Beam Ths example looks at the dynamc response of a cantlever beam n response to a cyclc force at the free end.
More informationMultiresolutional Techniques in Finite Element Method Solution of Eigenvalue Problem
Multresolutonal Technques n Fnte Element Method Soluton of Egenvalue Problem Marcn Kamńs Char of Mechancs of Materals, Techncal Unversty of Łódź Al. Poltechn 6, 93-59 Łodz, POLAND, tel/fax 48-4-63355 marcn@mm-lx.p.lodz.pl,
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 informationTime-Varying Systems and Computations Lecture 6
Tme-Varyng Systems and Computatons Lecture 6 Klaus Depold 14. Januar 2014 The Kalman Flter The Kalman estmaton flter attempts to estmate the actual state of an unknown dscrete dynamcal system, gven nosy
More informationFastener Modeling for Joining Composite Parts
AM-VPD09-006 Fastener Modelng for Jonng Composte Parts Alexander Rutman, Assocate Techncal Fellow, Sprt AeroSystems Chrs Boshers, Stress ngneer, Sprt AeroSystems John Parady, Prncpal Applcaton ngneer,
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 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 informationAmplification and Relaxation of Electron Spin Polarization in Semiconductor Devices
Amplfcaton and Relaxaton of Electron Spn Polarzaton n Semconductor Devces Yury V. Pershn and Vladmr Prvman Center for Quantum Devce Technology, Clarkson Unversty, Potsdam, New York 13699-570, USA Spn Relaxaton
More informationTransactions of the VŠB Technical University of Ostrava, Mechanical Series. article No. 1907
Transactons of the VŠB Techncal Unversty of Ostrava, Mechancal Seres No., 0, vol. LVIII artcle No. 907 Marek NIKODÝM *, Karel FYDÝŠEK ** FINITE DIFFEENCE METHOD USED FO THE BEAMS ON ELASTIC FOUNDATION
More informationUncertainty in measurements of power and energy on power networks
Uncertanty n measurements of power and energy on power networks E. Manov, N. Kolev Department of Measurement and Instrumentaton, Techncal Unversty Sofa, bul. Klment Ohrdsk No8, bl., 000 Sofa, Bulgara Tel./fax:
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 information