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A bit more about The Domain Reduction Method Boris Jeremić Professor, University of California, Davis Faculty Scientist, Lawrence Berkeley National Laboratory, Berkeley Ottawa Workshop 2010 Jeremić A bit more about The Domain Reduction Method The Domain Reduction Method (DRM) I Work by Bielak et al. (2003) at CMU. I Features: I I I I General 3D seismic input (p, s, Love, Rayleigh waves...) Nonlinear (elastic-plastic) ESSI Minimal outgoing waves, only radiation of NPPSSS energy Consistent replacement for seismic moment released from hypocenter with forces on a single layer of element around NPPSSS Jeremić A bit more about The Domain Reduction Method DRM I I I A large physical domain is to be analyzed for dynamic behavior. The source of disturbance is a known time history of a force field Pe (t). That source of loading is far away from a local feature which is dynamically excited by Pe (t) Local feature Ω ui ub Pe (t) Ω+ Γ ue Large scale domain Jeremić A bit more about The Domain Reduction Method DRM I I I I I remove local feature, create a free field model domain inside the boundary Γ is named Ω0 . outside boundary Γ, is Ω+ . outside domain Ω+ is the same as in the original model, simplification, is done on the domain inside boundary Γ. u0i Ω0 u0b Pe (t) Ω + Γ u0e Simplified large scale domain Jeremić A bit more about The Domain Reduction Method DRM I Equations of motions for a complete system M ü Jeremić A bit more about The Domain Reduction Method + K u = Pe DRM I Eq of motions written for each subdomain (interior, boundary and exterior of Γ) MiiΩ MibΩ 0 üi M Ω M Ω + M Ω+ M Ω+ ü + bi bb bb be b Ω+ Ω+ üe 0 Meb Mee Ω Kib 0 ui 0 Ω + K Ω+ K Ω+ u 0 Kbb = bb be b Ω+ Ω+ ue Pe Keb Kee KiiΩ KΩ bi 0 Jeremić A bit more about The Domain Reduction Method DRM Separate previous equation into two domains Ω and Ω+ MiiΩ MibΩ Ω MbiΩ Mbb Ω+ Ω+ Mbb Mbe Ω+ Ω+ Meb Mee üi üb + KiiΩ KibΩ Ω KbiΩ Kbb üb üe + Ω+ Ω+ Kbb Kbe Ω+ Ω+ Keb Kee ui ub = ub ue For this separation to work one needs to enforce I compatibility of displacements I equilibrium (through action–reaction forces Pb ) Jeremić A bit more about The Domain Reduction Method 0 Pb = ,Ω −Pb Pe , DRM I Simplified interior domain without local feature (with ui0 , ub0 , ue0 and Pb0 ) I The equations of motion in Ω+ for the auxiliary problem can now be written as: Ω+ Ω+ Mbb Mbe Ω+ Ω+ Meb Mee üb0 üe0 Jeremić A bit more about The Domain Reduction Method + Ω+ Ω+ Kbb Kbe Ω+ Ω+ Keb Kee ub0 ue0 = −Pb0 Pe DRM Use second part of previous equation to obtain the dynamic force Pe as Ω+ 0 Ω+ 0 Ω+ 0 Ω+ 0 Pe = Meb üb + Mee üe + Keb ub + Kee ue Jeremić A bit more about The Domain Reduction Method DRM The total displacement, ue , can be expressed as the sum of the I free field ue0 (from the background, simplified model, free field), and I the residual field we (comming from the local feature) ue = ue0 + we I this is a change of variables and not superposition! Jeremić A bit more about The Domain Reduction Method DRM By substitution into full equation one gets MiiΩ MibΩ 0 üi M Ω M Ω + M Ω+ M Ω+ ü + bi bb bb be 0 b Ω+ Ω+ üe + ẅe 0 Meb Mee KibΩ 0 ui 0 Ω + K Ω+ K Ω+ u 0 Kbb = bb be 0 b Ω+ Ω+ P u + w Keb Kee e e e KiiΩ KΩ bi 0 Jeremić A bit more about The Domain Reduction Method DRM After moving the free field motions ue0 to the right hand side, prevous equation becomes MiiΩ MibΩ 0 üi M Ω M Ω + M Ω+ M Ω+ ü + bi bb bb be b Ω+ Ω+ ẅ 0 Meb Mee e Ω Ω Kii Kib 0 ui K Ω K Ω + K Ω+ K Ω+ u = bi bb bb be b Ω+ Ω+ we 0 Keb Kee 0 Ω+ 0 Ω+ 0 −Mbe üe − Kbe ue Ω+ 0 Ω+ 0 −Mee üe − Kee ue + Pe Jeremić A bit more about The Domain Reduction Method DRM By substitution of Pe MiiΩ MibΩ 0 üi M Ω M Ω + M Ω+ M Ω+ ü + bi bb be bb b Ω+ Ω+ ẅ 0 Meb Mee e Ω Ω Kii Kib 0 ui K Ω K Ω + K Ω+ K Ω+ u = bi bb bb be b Ω+ Ω+ we 0 Keb Kee 0 Ω+ 0 Ω+ 0 −Mbe üe − Kbe ue Ω+ 0 Ω+ 0 Meb üb + Keb ub Jeremić A bit more about The Domain Reduction Method DRM The right hand side is the dynamically consistent replacement force (so called effective force, P eff ) for the dynamic source forces Pe . P eff eff 0 Pi Ω+ 0 Ω+ 0 −Mbe üe − Kbe ue = = Pbeff eff Ω+ 0 Ω+ 0 Pe Meb üb + Keb ub Jeremić A bit more about The Domain Reduction Method DRM I Seismic forces Pe replaced by P eff I P eff applied only to a single layer of elements next to Γ. I Only outgoing waves from dynamics of the NPP I Material inside Ω can be elastic-plastic Ω Γe A bit more about The Domain Reduction Method ue + Ω Pe Jeremić Γ ub Fault ui ue Γ +
