By Ulrich Römer
This ebook offers a entire mathematical method for fixing stochastic magnetic box difficulties. It discusses variability in fabric homes and geometry, with an emphasis at the renovation of structural actual and mathematical houses. It particularly addresses uncertainties within the desktop simulation of magnetic fields originating from the producing technique. Uncertainties are quantified via approximating a stochastic reformulation of the governing partial differential equation, demonstrating how records of actual amounts of curiosity, resembling Fourier harmonics in accelerator magnets, can be utilized to accomplish strong designs. The booklet covers a couple of key tools and effects equivalent to: a stochastic version of the geometry and fabric homes of magnetic units in keeping with dimension information; a close description of numerical algorithms in line with sensitivities or on a higher-order collocation; an research of convergence and potency; and the applying of the built version and algorithms to uncertainty quantification within the advanced magnet structures utilized in particle accelerators.
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Extra resources for Numerical Approximation of the Magnetoquasistatic Model with Uncertainties: Applications in Magnet Design
Sample text
However, for the case ν˜ = νL,0 this is guaranteed as νL,0 > 0 holds. 1 (p-Laplace) For illustration, we consider a well-known two-dimensional and static example originally proposed in [1], which in the present form is adapted from [30]. Let the computational domain be D = DC = (0, 1) × (0, 1). 81) is a frequently used prototype for quasilinear elliptic models [39]. In [40], we have studied this example in a stochastic setting by modeling p as a random parameter. We assume homogeneous Dirichlet boundary conditions and impose a constant excitation J = 2.
For a precise definition of B-splines see Appendix B. 33d) see [25]. 34) respectively, where in turn DF refers to the Jacobian of the transformation F and det(DF) to its determinant. 3 Finite Element Formulation Given discrete spaces, a finite element formulation can be derived in a straightforward way by simply substituting them for their continuous counterparts. 14) and postpone the technically more involved time-transient case. 3). Then the finite element formulation reads, find Ah ∈ Wst,h (D) such that a(Ah ; vh ) = l(vh ), ∀vh ∈ Wst,h (D).
Methods Appl. Mech. Eng. 196(1), 210–224 (2006) 2. : Numerical analysis of nonlinear multiharmonic eddy current problems. Numerische Mathematik 100(4), 593–616 (2005) 3. : Sobolev spaces, vol. 140. Academic Press (2003) References 37 4. : Numerical analysis of nonlinear multiharmonic eddy current problems. Technical report, SFB Numerical and Symbolic Scientific Computing, Johannes Kepler University Linz, Austria (2004) 5. : Existence and uniqueness of eddy current problems in bounded and unbounded domains.