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Parametric modeling and model order reduction for (electro-)thermal analysis of nanoelectronic structures
- Lihong Feng^{1},
- Yao Yue^{1}Email authorView ORCID ID profile,
- Nicodemus Banagaaya^{1},
- Peter Meuris^{2},
- Wim Schoenmaker^{2} and
- Peter Benner^{1}
https://doi.org/10.1186/s13362-016-0030-8
© Feng et al. 2016
Received: 14 February 2016
Accepted: 18 October 2016
Published: 8 November 2016
Abstract
In this work, we discuss the parametric modeling for the (electro)-thermal analysis of components of nanoelectronic structures and automatic model order reduction of the consequent parametric models. Given the system matrices at different values of the parameters, we introduce a simple method of extracting system matrices which are independent of the parameters, so that parametric models of a class of linear parametric problems can be constructed. Then the reduced-order models of the large-scale parametric models are automatically obtained using a posteriori output error bounds for the reduced-order models. Simulations of both thermal and electro-thermal systems confirm the validity of the proposed methods.
Keywords
- Posteriori Error
- Expansion Point
- Reduce Basis Method
- Empirical Interpolation Method
- Parametric Model Order Reduction
1 Introduction
Parameter variations have become essential in the design of micro- and nano-electronic (-mechanical) systems as well as of coupled electro-thermal problems, since in many analyses such as optimization and uncertainty quantification, modeling and simulation at many values of the parameters are unavoidable. For many design and analysis tools, modeling and simulation need to be done at each instance of the parameter from scratch: given a fixed value of the parameter, say \(p^{*}\), a certain numerical discretization method, e.g., a finite element method, is used to build a spatially discretized model only valid for \(p^{*}\), and numerical integration is then performed to get the output response corresponding to \(p^{*}\). If additional analysis beyond the capability of the aforementioned software is required, the software often can provide only the (conductivity, capacitance) matrices corresponding to certain samples of the parameter, rather than explicit matrix functions that are more convenient for mathematical analysis.
It is desired to derive a single parametric discretized system that is valid for all possible values of the parameters, so that discretization does not have to be implemented anew for each value of interest, which can save much simulation time. In this paper, we propose a simple method of extracting matrix functions that is capable of calculating the matrices corresponding to any parameter value efficiently. Thanks to these matrix functions, the dynamics of the parametric system can be described by a single system of parametric ordinary differential equations (ODEs) or differential-algebraic equations (DAEs). The approach is in particular suitable for the (electro)-thermal analysis of nanoelectronic structures, as the parameters there often appear in a linear affine form required by this extraction for a parametric model.
Simulating the consequent parametric system is, however, still very time consuming, because of the high dimension of the system. We propose to use parametric model order reduction (PMOR) to compute a reduced-order model (ROM) that is not only of a much lower dimension, but also accurate for all values of the parameters within a specified range. Therefore, using the parametric ROM to replace the full-order model (FOM) in simulation and other analyses like optimization and uncertainty quantification leads to significant speedup and high accuracy. Many PMOR methods have been proposed so far. A survey of PMOR methods can be found in [1]. In this paper, we use a multi-moment-matching PMOR method [2] to construct the reduced-order model. These methods are popular in practical applications since they are easy to implement, need less computations than most of the other methods, and are therefore suitable to reduce high-dimensional ODE/DAE systems that commonly arise in design and analysis of VLSI (very-large-scale integration) circuits. Furthermore, we propose to use an a posteriori output error bound [3] to construct the ROM automatically, i.e., the algorithm can build a reduced-order model satisfying a prescribed error tolerance without further specification of algorithmic parameters, e.g., interpolation points and the order of the ROM, which can be automatically determined by the algorithm in an adaptive manner.
The paper is organized as follows. In Section 2, we propose a simple method of extracting the state-space representation of a class of parametric problems. Section 3 reviews the basic idea of PMOR methods and Section 4 describes an algorithm that implements the multi-moment-matching PMOR method adaptively based on an a posteriori output error bound for the ROM. Section 5 describes the (electro-)thermal simulation for two test models: a package model and a Power-MOS device model. The parametric modeling and PMOR of these models, especially the extraction of the tensors and PMOR for the one-way nonlinearly coupled dynamical system, are discussed in Section 6. The numerical results are presented in Section 7, and the paper is concluded in Section 8. In all test cases, the matrices are efficiently extracted and the parametric ROMs automatically obtained meet the requirements on accuracy and compactness.
2 Parametric modeling
Simulating the system in (4) may still take a lot of time when the dimension n is large, especially when it has to be simulated at many samples of p (like in optimization). In the next section, we propose to use PMOR to construct a parametric reduced-order model, which will replace the original large-scale system in (4) in simulations for speedup. Since the size of the reduced-order model is usually much smaller than n, simulation can be conducted within a much shorter and more reasonable time period.
3 PMOR based on multi-moment-matching
The choice of the number and locations of the expansion points \((s_{i},p^{i})\) has an important influence on the efficiency of multi-moment-matching PMOR methods. Actually, good accuracy and compactness of the reduced-order model can only be achieved when the expansion points are selected judiciously.
In the next section, we introduce a technique for adaptively selecting the expansion points according to an a posteriori error bound \(\Delta(s,p)\) for the ROM. By using the error bound to access the reliability of the reduced-order model, we develop an automatic procedure for constructing the ROM.
4 Adaptively selecting the expansion points
5 Test models
• The thermal-only option takes Q as an independent input. Under this option, the electrical sub-system and the thermal sub-system are completely decoupled. The thermal-only option is especially interesting to the package model, since it can be used to study the thermal dynamics stimulated by heat-injecting or extracting properties on the boundary of the simulation domain, Joule self-heating, etc. Finite element discretization of the thermal-only option leads to a linear dynamical system exactly the form (2).
6 Parametric modeling and PMOR for the test models
Note that the method developed in Section 2 also applies to the tensor F due to the following reasoning. Every slice of F, say \(F_{i}(p) \in\mathbb{R}^{n_{\mathrm{E}} \times n_{\mathrm{E}}}\), can be extracted using the procedure from (6) to (8), and under the same set of samples \(p^{a_{1}}, p^{a_{2}}, \ldots, p^{a_{m}}\) in (6), the obtained coefficients \(\tilde{p}_{11}\), \(\tilde{p}_{12}\), …, \(\tilde{p}_{mm}\) in (8) are the same. Therefore, the computation for all slices can be conducted together in the tensor form, i.e., assuming that \(M_{1}, M_{2}, \ldots, M_{m}\) in (6) are tensors, they can be extracted by (8), using the coefficient computed by (9).
Therefore, to extract all matrices and tensors for the package model, we need first only to invert a single \(3\times3\) matrix, and then, for each matrix or tensor function, we need to calculate (8) once.
The remaining problem is how to reduce system (17b), which has a quadratic one-way coupling term. To simplify the presentation, we use the Power-MOS circuit with the parameter dependence \(M(\sigma)=M_{0}+\sigma M_{1}\) as an example. Following the idea presented in [12], we first ignore the nonlinear part \(F(p) \times_{2} x_{\mathrm{E}}(p) \times_{3} x_{\mathrm{E}}(p)\) in system (17b) and use the adaptive PMOR algorithm proposed to reduce the resulting system in the form (4) [7, 9]. To approximate the one-way coupling term, we need to reduce the electrical sub-system before the thermal sub-system.
• The electrical sub-system (17a) is already in the form (10) if we assign \(E(p)=0\), \(A(p)=-A_{\mathrm{E}}(p)\), \(B(p)=-B_{\mathrm{E}}(p)\), \(s=t\), by noting that for the validity of the proposed PMOR method, system (10) is actually not necessarily a frequency-domain system. Denote the basis built for the electrical sub-system (17a) by \(V_{\mathrm{E}}\). For MOR for algebraic equations, it is worth mentioning the exact reduction method proposed in [13] for non-parametric systems, which does not require an error bound. However, the method we propose in this paper can not only reduce parametric systems, but also normally build a reduced-order model of a much lower dimension.
7 Numerical results
In this section, we first show the numerical results of the thermal analysis of the package model. Then, we present the numerical results of the electro-thermal analysis of both the package model and the Power-MOS device model.
7.1 Numerical results for the thermal analysis
\(\pmb{V_{ s_{i}, p^{i}}=\operatorname{span}\{R_{0}, R_{1} \}_{ s_{i}, p^{i}}}\) , \(\pmb{i=1,2}\) , \(\pmb{\varepsilon_{\mathrm{tol}}=10^{-3}}\) , \(\pmb{n=8,549}\) , \(\pmb{r=58}\)
Iteration i | \(\boldsymbol {(s_{0}, h^{i})}\) | \(\boldsymbol {\Delta(s_{0}, h^{i})}\) |
---|---|---|
1 | (200πȷ,0.3834) | 0.0153 |
2 | (200πȷ,0.0677) | 5 × 10^{−4} |
7.2 Numerical results for the electro-thermal analysis
First, we apply the matrix extraction algorithm and adaptive PMOR method developed to the electro-thermal simulation of the package model with 34 inputs and 68 outputs.
Convergence behavior of electro-thermal simulation of the package model ( \(\pmb{\epsilon_{\mathrm{tol}}=10^{-4}}\) )
Iteration | Electrical sub-system | Thermal sub-system | ||
---|---|---|---|---|
Selected sample h | Error bound | Selected sample ( s , h ) | Error bound | |
1 | 1 | 2.1 × 10^{3} | (8.1339,7.5910) | 7.3 × 10^{6} |
2 | 100 | 3.7 × 10^{0} | (41.065,29.653) | 2.3 × 10^{1} |
3 | 90 | 6.6 × 10^{−2} | (17.494,15.121) | 1.3 × 10^{−1} |
4 | 80 | 6.4 × 10^{−3} | (16.455,4.6942) | 7.8 × 10^{−5} |
5 | 70 | 5.3 × 10^{−3} | – | – |
6 | 60 | 4.2 × 10^{−3} | – | – |
7 | 50 | 3.1 × 10^{−3} | – | – |
8 | 40 | 1.8 × 10^{−3} | – | – |
9 | 30 | 8.9 × 10^{−4} | – | – |
Convergence behavior of electro-thermal simulation of the Power-MOS model ( \(\pmb{\epsilon_{\mathrm{tol}}=10^{-12}}\) )
Iteration | Electrical sub-system | Thermal sub-system | ||
---|---|---|---|---|
Selected sample σ | Error bound | Selected sample ( s , σ ) | Error bound | |
1 | 10^{7} | 7.165399 × 10^{−24} | (0,2.736 × 10^{7}) | 43.73 |
2 | – | – | (10^{6},2.537 × 10^{7}) | 4.225 × 10^{−4} |
3 | – | – | (2.632 × 10^{5},1.694 × 10^{7}) | 4.345 × 10^{−8} |
4 | – | – | (5.790 × 10^{5},2.687 × 10^{7}) | 9.774 × 10^{−11} |
5 | – | – | (5.263 × 10^{4},2.836 × 10^{7}) | 4.041 × 10^{−13} |
Figure 4(b) shows that the relative error is large when t is small, e.g., with a value in the range of \([10,100]\) at the time \(10^{-9}\ \mbox{s}\). The reason is that the thermal flux is still very close to zero (the circuit is hardly heated up) and the numerical error arising from the discretization of the FOM results in numerical noise, which dominates the output of the FOM when the true physical dynamics is small. As Figure 4(b) shows, the ROM approximates the thermal flux accurately after the thermal flux dominates the numerical error (\(t > 2 \times10^{-7}\)). Therefore, the ROM can not only approximate the true dynamics accurately, but is also robust to the numerical error present in the FOM due to discretization. Furthermore, although the samples are selected within the range \([10^{7}, 5 \times10^{7}]\), Figure 4(b) shows that the parametric ROM is valid in a much wider range.
8 Conclusions and further discussion
We have proposed a simple automatic matrix extracting technique for a class of parametric dynamical systems, and shown that automatic parametric model order reduction can be realized with the guidance of an a posteriori error bound. The above techniques have been successfully applied to the thermal simulation of a package model, and the electro-thermal simulation of a package model and a Power-MOS device model. Compact and reliable reduced-order models have been automatically obtained, which offers the possibility of being integrated into dedicated electro-thermal simulation software to accelerate design automation.
It is worth pointing out that although the adaptive Algorithm 1 for multi-moment-matching PMOR methods resembles the greedy algorithm that is often used in the reduced basis method [6], the size of the training sets we used in numerical tests, which is 20 in all three examples, is much smaller than that typically used in reduced basis methods, which can easily reach 1,000 or even 10,000. Numerical simulations show that this small number of training points leads to accurate ROMs within a large parameter range. Another phenomenon we observed in numerical tests for electro-thermal analysis is that the resulting parametric ROMs are robust to numerical error introduced by PDE discretization.
Declarations
Acknowledgements
This work is financially supported by the collaborative project nanoCOPS [14], Nanoelectronic COupled Problems Solutions, supported by the European Union in the FP7-ICT-2013-11 Program under Grant Agreement Number 619166.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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