###### Abstract  
In simulations of three-dimensional transient physics filled through a numerical approach, the order of the equation set of high-fidelity models is extremely high. To eliminate the large dimension of equations, a model order reduction (MOR) technique is introduced. In the existing MOR methods, the block Arnoldi algorithm-based MOR method is numerically stable, achieving a passively reduced order model. Nevertheless, this method performs poorly when it is applied to very wide-frequency transients. To eliminate this deficiency, multipoint MOR methods are emerging. However, it is hard to directly apply an existing multipoint MOR method to a 3-D transient field equation set. The implementation issues in a reduction process (such as the selection of expansion points, the number of moments matched at a point and the error bound) have not been explored in detail. In this respect, an adaptive multipoint model reduction model based on the Arnoldi algorithm is proposed to obtain the reduced-order models of a 3-D temperature field. The originality of this study is the proposal of a novel adaptive algorithm for selecting expansion points, matching moments automatically, using a posterior-error estimator based on temperature response coupled with a network topological method (NTM). The computational efficiency and accuracy of the proposed method are evaluated by the numerical results from solving the temperature field of a prototype insulated-gate bipolar transistor (IGBT).  
model order reduction (MOR); 3-D transient field problems; broad-band frequency; moment matching; multi-point Article  
A Block Arnoldi Algorithm Based Reduced-Order Model Applied to Large-Scale Algebraic Equations of a 3-D Field  
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MOR  
A Block Arnoldi Algorithm Based Reduced-Order Model Applied to Large-Scale  
example, the Asymptotic Waveform Evaluation (AWE), as reported in [8], has focused on solving the number and generating mechanism of the expansion points by considering complex frequency hopping (CFH). Authors in [9; 10; 11] have focused on the error bound and selection of expansion points for Pade-via-Lanczos (PVL). In addition, authors in [12] have focused on the error bound and selection of expansion points for Efficient Nodal Order Reduction (ENOR) by embedding proper orthogonal decomposition (POD).  
However, both AWE and ENOR use a recursive expression to explicitly compute the moments of the high-fidelity model. Therefore, AWE and ENOR are not numerically stable [13]. Although PVL is a powerful and efficient algorithm, it cannot preserve some physically feasible conditions such as model passivity [13]. On the other hand, in the MOR methods based on moment matching, the Passive Reduced-order Interconnect Macromodeling Algorithms (PRIMA), based on the Arnoldi method, are numerically stable and can preserve the passivity of the high-fidelity model in RLC circuits. Nevertheless, this method is difficult to directly employ on a high-fidelity model of a three-dimensional transient field since this methodology is intended to deal with interconnect circuit equations, which are different in form from 3-D transient field equations. Furthermore, the difficulty of applications in broadband situations for PRIMA has not been explored in detail in the literature. In this regard, we propose an adaptive multipoint model reduction model based on the Arnoldi algorithm to obtain the reduced-order models of a 3-D temperature field.