44.3 Numerical methods Numerical solutions for the field equations are now routine for a large number of problems encountered in magnet design; these include, for example, two-dimensional models taking into account nonlinear, anisotropi and even hysteretic effects. Their use for complete three-dimensional models is not so widespread because of ne escalating computer resources needed as the problem size increases. Nevertheless, 3-D solutions for non- linear statics devices are regularly obtained in industry, and time-dependent solutions are rapidly becoming more cost effective as computer hardware architectures develop Finite elements This increasing use of computer-based solutions has come about largely because of the generality of the finite element method(FEM). In this method, the problem space is subdivided(discretized) into finite regio (elements)over which the solution is assumed to follow a simple local approximating trial function(shap functions). In the simplest situation, a particular element could be a plane hexahedra defined by its eight vertices or nodes and a solution of Eq (44.3)may be approximated by a, a,+ a3y+a,2+asxy a yz+ a,zx t a NU;(44.5 Because a hexahedra has eight nodes it is natural to select a bilinear trial function with eight parameters; see Fig. 44.1 for other examples. The functions N, are called the local shape functions and the parameters U, are lements can be integrated into a numerical model for the whole problem space either by(a) the variational method in which the total energy of the system is expressed in terms of the finite element trial functions and then minimized to determine the best solution or(b)the weighted residual method in which the formal error(residual), arising by substituting the trial functions into the defining equation, is weighted by a suitably chosen function and then integrated over the problem domain. The best fit for the trial function parameters can then be obtained by equating the integral to zero. Both methods lead to a set of algebraic equations and are equivalent if the weighting functions are chosen to be the trial functions Galerkins method [Zienkiewicz, 1990)). At the element level, the residual R, is given by VNμVNd92U1+|NQd (44.6) clem where Q(RHS)denotes the sources of Eqs. (44.2)or(44. 3). The integrals can be readily evaluated and assembled for the whole problem by superposition, taking account of the boundary conditions and removing the redun ancy at shared nodes. At interelement boundaries in a region of particular potential [reduced Eq (44. 2)or total Eq(44.3)] the solution is forced to be continuous, but the normal flux (i.e. uaU/an)will only be continuous in a weak sense, that is to say the discontinuity will depend upon the mesh refinement. The FEM provides a systematic technique for replacing the continuum field equations with a set of discrete algebraic equations that can be solved by standard procedures. In Fig. 44. 2 a typical field map is shown for a permanent magnet machine modeled by a computer simulator that can take into account nonlinearity and permanently magnetized materials. Although hysteresis effects can be included, the computational resources equired can be prohibitive because of the vector nature of magnetization. The magnetic material must be haracterized by a large number of measurements to take account of the minor loops, and from these the convolution integrals necessary to obtain the constitutive relationships can be evaluated [Mayergoyz, 1990] These characteristics must then be followed through time; this can be implemented by solving at a discrete set of time points, given the initial conditions in the material c 2000 by CRC Press LLC© 2000 by CRC Press LLC 44.3 Numerical Methods Numerical solutions for the field equations are now routine for a large number of problems encountered in magnet design; these include, for example, two-dimensional models taking into account nonlinear, anisotropic, and even hysteretic effects. Their use for complete three-dimensional models is not so widespread because of the escalating computer resources needed as the problem size increases. Nevertheless, 3-D solutions for non￾linear statics devices are regularly obtained in industry, and time-dependent solutions are rapidly becoming more cost effective as computer hardware architectures develop. Finite Elements This increasing use of computer-based solutions has come about largely because of the generality of the finite element method (FEM). In this method, the problem space is subdivided (discretized) into finite regions (elements) over which the solution is assumed to follow a simple local approximating trial function (shape functions). In the simplest situation, a particular element could be a plane hexahedra defined by its eight vertices or nodes and a solution of Eq. (44.3) may be approximated by (44.5) Because a hexahedra has eight nodes it is natural to select a bilinear trial function with eight parameters; see Fig. 44.1 for other examples. The functions Ni are called the local shape functions and the parameters Ui are the solution values at the nodes. The finite elements can be integrated into a numerical model for the whole problem space either by (a) the variational method in which the total energy of the system is expressed in terms of the finite element trial functions and then minimized to determine the best solution or (b) the weighted residual method in which the formal error (residual), arising by substituting the trial functions into the defining equation, is weighted by a suitably chosen function and then integrated over the problem domain. The best fit for the trial function parameters can then be obtained by equating the integral to zero. Both methods lead to a set of algebraic equations and are equivalent if the weighting functions are chosen to be the trial functions (Galerkin’s method [Zienkiewicz, 1990]). At the element level, the residual Ri is given by (44.6) where Q (RHS) denotes the sources of Eqs. (44.2) or (44.3). The integrals can be readily evaluated and assembled for the whole problem by superposition, taking account of the boundary conditions and removing the redun￾dancy at shared nodes. At interelement boundaries in a region of particular potential [reduced Eq. (44.2) or total Eq. (44.3)] the solution is forced to be continuous, but the normal flux (i.e., m]U/]n) will only be continuous in a weak sense, that is to say the discontinuity will depend upon the mesh refinement. The FEM provides a systematic technique for replacing the continuum field equations with a set of discrete algebraic equations that can be solved by standard procedures. In Fig. 44.2 a typical field map is shown for a permanent magnet machine modeled by a computer simulator that can take into account nonlinearity and permanently magnetized materials. Although hysteresis effects can be included, the computational resources required can be prohibitive because of the vector nature of magnetization. The magnetic material must be characterized by a large number of measurements to take account of the minor loops, and from these the convolution integrals necessary to obtain the constitutive relationships can be evaluated [Mayergoyz, 1990]. These characteristics must then be followed through time; this can be implemented by solving at a discrete set of time points, given the initial conditions in the material. y ª u = a1 2 + a x + a3 y + a4z + a5 xy + a6 yz + a7zx + a8 xyz = Â Ni Ui R N N d U N Qd i = — i — j j i È Î Í Í ˘ ˚ ˙ ˙ + Ú Ú m W W elem elem