LEADER 01659oam 2200493K 450 001 9910712909203321 005 20260430112634.0 024 8 $aGOVPUB-C13-baa5e93e3ff82a0446b99c629e97550c 035 $a(CKB)5470000002499105 035 $a(OCoLC)629721430$z(OCoLC)664529230$z(OCoLC)681180109 035 $a(OCoLC)995470000002499105 035 $a(DGPO)001116530 035 $a(EXLCZ)995470000002499105 100 $a20100523d1965 ua 0 101 0 $aeng 135 $aurbn||||||abp 135 $aurbn||||||ada 181 $ctxt$2rdacontent 182 $cc$2rdamedia 183 $acr$2rdacarrier 200 10$aSpot diagrams for the prediction of lens performance from design data /$fOrestes N. Stavroudis and Loyd E. Sutton 210 1$aWashington, D.C. :$cU.S. Dept. of the Commerce, National Bureau of Standards :$cG.P.O.,$d1965. 215 $a1 online resource (iii, 96 pages) $cillustrations 225 1 $aNBS monograph ;$v93 320 $aIncludes bibliographical references (page 15).$b20 606 $aPhotographic lenses 606 $aPhotographic lenses$2fast 615 0$aPhotographic lenses. 615 7$aPhotographic lenses. 676 $a681.423 700 $aStavroudis$b O. N$g(Orestes Nicholas),$f1923-$027009 702 $aSutton$b Loyd E. 712 02$aUnited States.$bNational Bureau of Standards. 801 0$bOCLCE 801 1$bOCLCE 801 2$bAZU 801 2$bOCLCQ 801 2$bOCLCO 801 2$bOCLCF 801 2$bOCLCQ 906 $aBOOK 912 $a9910712909203321 996 $aSpot diagrams for the prediction of lens performance from design data$93447616 997 $aUNINA LEADER 04914nam 22005895 450 001 9910558492603321 005 20260605205333.0 010 $a9783030981914$b(electronic bk.) 024 7 $a10.1007/978-3-030-98191-4 035 $a(MiAaPQ)EBC6944417 035 $a(Au-PeEL)EBL6944417 035 $a(CKB)21459774800041 035 $a(PPN)261520202 035 $a(OCoLC)1309070124 035 $a(DE-He213)978-3-030-98191-4 035 $a(EXLCZ)9921459774800041 100 $a20220331d2022 u| 0 101 0 $aeng 135 $aurcnu|||||||| 181 $ctxt$2rdacontent 182 $cc$2rdamedia 183 $acr$2rdacarrier 200 10$aOptimal Design of Multi-Phase Materials $eWith a Cost Functional That Depends Nonlinearly on The Gradient /$fby Juan Casado-Díaz 205 $a1st ed. 2022. 210 1$aCham :$cSpringer International Publishing :$cImprint: Springer,$d2022. 215 $a1 online resource (119 pages) 225 1 $aSpringerBriefs in Mathematics,$x2191-8201 311 08$aPrint version: Casado-Díaz, Juan Optimal Design of Multi-Phase Materials Cham : Springer International Publishing AG,c2022 9783030981907 320 $aIncludes bibliographical references. 327 $aChapter 1. Homogenization of Elliptic PDE with Varying Coefficients -- Chapter 2. The Relaxed Formulation of an Optimal Design Problem via Homogenization Theory -- Chapter 3. Optimality Conditions and Numerical Resolution -- Chapter 4. Some Extesions: Multi-State and Evolutive Problems. 330 $aThis book aims the optimal design of a material (thermic or electrical) obtained as the mixture of a finite number of original materials, not necessarily isotropic. The problem is to place these materials in such a way that the solution of the corresponding state equation minimizes a certain functional that can depend nonlinearly on the gradient of the state function. This is the main novelty in the book. It is well known that this type of problems has no solution in general and therefore that it is needed to work with a relaxed formulation. The main results in the book refer to how to obtain such formulation, the optimality conditions, and the numerical computation of the solutions. In the case of functionals that do not depend on the gradient of the state equation, it is known that a relaxed formulation consists of replacing the original materials with more general materials obtained via homogenization. This includes materials with different properties of the originals but whose behavior can be approximated by microscopic mixtures of them. In the case of a cost functional depending nonlinearly on the gradient, it is also necessary to extend the cost functional to the set of these more general materials. In general, we do not dispose of an explicit representation, and then, to numerically solve the problem, it is necessary to design strategies that allow the functional to be replaced by upper or lower approximations. The book is divided in four chapters. The first is devoted to recalling some classical results related to the homogenization of a sequence of linear elliptic partial differential problems. In the second one, we define the control problem that we are mainly interested in solving in the book. We obtain a relaxed formulation and their main properties, including an explicit representation of the new cost functional, at least in the boundary of its domain. In the third chapter, we study the optimality conditions of therelaxed problem, and we describe some algorithms to numerically solve the problem. We also provide some numerical experiments carried out using such algorithms. Finally, the fourth chapter is devoted to briefly describe some extensions of the results obtained in Chapters 2 and 3 to the case of dealing with several state equations and the case of evolutive problems. The problems covered in the book are interesting for mathematicians and engineers whose work is related to mathematical modeling and the numerical resolution of optimal design problems in material sciences. The contents extend some previous results obtained by the author in collaboration with other colleagues. 410 0$aSpringerBriefs in Mathematics,$x2191-8201 606 $aMathematical analysis 606 $aMathematics 606 $aAnalysis 606 $aApplications of Mathematics 606 $aEquacions en derivades parcials$2thub 608 $aLlibres electrònics$2thub 615 0$aMathematical analysis. 615 0$aMathematics. 615 14$aAnalysis. 615 24$aApplications of Mathematics. 615 7$aEquacions en derivades parcials 676 $a515.35 676 $a620.118 700 $aCasado-Di?az$b Juan$01220448 801 0$bMiAaPQ 801 1$bMiAaPQ 801 2$bMiAaPQ 912 $a9910558492603321 996 $aOptimal Design of Multi-Phase Materials$92824627 997 $aUNINA