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Parametric modeling and model order reduction for (electro-)thermal analysis of nanoelectronic structures


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.

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

In this section, we introduce a method for extracting system matrices of a class of parametric problems, so that the parametric representation of the models in state-space form can be derived. Assume that the parametric problem can be generally described by the following partial differential equation,

$$ \frac{\partial a(t,z;p)}{\partial t}+\mathcal {L}\bigl[a(t,z;p)\bigr]=f(t,z;p),\quad t \in[0,T], z\in\Omega, p\in\mathcal {P}, $$

where \(\mathcal {L}[\cdot]\) is a linear spatial differential operator, \(f(t,z;p)\) is the excitation, \(p=(p_{1},\ldots,p_{m})^{T}\) is a vector of parameters, \(\Omega\subseteq \mathbb {R}^{d} \) (\(d=1,2,3\)) is the spatial domain and \(\mathcal {P}\subseteq \mathbb {R}^{m}\) is the parameter domain.

In many engineering problems, we are interested in the input-output behavior of system (1). In simulation of integrated circuits, for example, we are often concerned with the currents at the contacts (outputs) rather than inside the circuit when certain voltages are exerted on the contacts (inputs). In such cases, state-space representation is often used for the spatially discretized system. Using finite-element simulation software, we can usually conduct spacial discretization only at a fixed value \(p^{*}\) of p and obtain the discretized system

$$ \begin{aligned} &E\bigl(p^{*}\bigr)\frac{dx(t, p^{*})}{dt}=A \bigl(p^{*}\bigr)x\bigl(t, p^{*}\bigr)+B\bigl(p^{*}\bigr)u(t), \\ &y\bigl(t, p^{*}\bigr)=C\bigl(p^{*}\bigr)x\bigl(t, p^{*}\bigr)+D\bigl(p^{*}\bigr)u(t), \end{aligned} $$

where only \(E(p^{*}), A(p^{*})\in\mathbb {R}^{n\times n}\), \(B(p^{*}) \in \mathbb {R}^{n \times l_{\mathrm{I}}}\), \(C(p^{*}) \in\mathbb {R}^{l_{\mathrm{O}}\times n}\), and \(D(p^{*}) \in\mathbb {R}^{l_{\mathrm{O}} \times l_{\mathrm{I}}}\) at the fixed value \(p^{*}\) of p are available. Here, \(x \in\mathbb {R}^{n}\) is the state vector, and \(y\in \mathbb {R}^{l_{O}}\) is the output response. For design purposes, the simulation results at many fixed values of p should be derived and analyzed. If we simply use the software, the discretization must be repeated at many values of p. To avoid repeated discretization in space, and hence to save design time, it is desired that a parametric representation of the model is available.

We will show that if \(E(p), A(p), B(p), C(p), D(p)\) are in the form of

$$ M(p)=p_{1} M_{1}+\cdots+p_{m} M_{m}, $$

we can easily compute \(M_{1}, \ldots, M_{m}\) based on the data of \(M(p)\) at m fixed values of p (here and below, \(M(p)\) stands for any of the matrices \(E(p), A(p), B(p), C(p), D(p)\)). Hence, the parametric representation of (2) is available, i.e.,

$$ \begin{aligned} &E(p)\frac{dx(t, p)}{dt}=A(p)x(t, p)+B(p)u(t), \\ &y(t, p)=C(p)x(t, p)+D(p)u(t). \end{aligned} $$

The discretized parametric model in (4) not only prevents repeated discretization at all values of p, but also retains the same system order n as the nonparametric system (2) regardless of the number of the parameters. Note that the linear-affine form (3) does not require linear dependence on geometrical and/or physical parameters since \(p_{i}\) may present abstract parameters. It covers a rather broad range of applications since any system of the form

$$ M(q)=\phi_{1}(q) M_{1} + \cdots+ \phi_{m}(q) M_{m} , $$

where \(\phi_{i}\) represents an arbitrary scalar function of q, can easily be rewritten into form (3) with the change of parameter \(p_{i}=\phi_{i}(q)\). The vectors p and q can be of different lengths, e.g., for the package model that will be described in Section 5, we set \(p_{1}=1\), \(p_{2}=h\), \(p_{3}=\frac {1}{h}\), where h is the top layer thickness of the package. Note that any parametric matrix \(M(q)\) can be written into the form (5) theoretically since if we denote the \((i,j)\)-th entry of \(M(q)\) by \(m_{i,j}(q)\), \(M(q)\) can be at least expanded as

$$M(q)=\sum_{i=1}^{n} \sum _{j=1}^{n}m_{i,j}(q) \bigl( e_{i} e_{j}^{\mathrm {T}} \bigr), $$

where \(e_{i}\) is the i-th column of the \(n \times n\) identity matrix. In practice, however, m in (5) should be small enough, e.g., below ten, if PMOR is expected to be efficient on the extracted system. A concise form can often be obtained by physical reasoning provided by engineers. For our test examples in Section 5, \(M(h)=M_{0} + h M_{1} + \frac{1}{h} M_{2}\) when the thickness parameter h changes, and \(M(\sigma)=M_{0} + \sigma M_{1}\) when the conductivity parameter σ changes. However, there are still cases where the space-discretization does not yield a concise affine-linear structure. For such cases, empirical interpolation method [1, 4] is commonly applied to obtain an approximation with an affine structure.

Suppose that m groups of matrices \(E(p^{a_{i}}), A(p^{a_{i}}), B(p^{a_{i}}), C(p^{a_{i}}), D(p^{a_{i}})\) have been obtained, e.g., by simulation software at m different samples \(p^{a_{i}}\), \(i=1,\ldots, m\). Using the formulation in (3), one can get a group of equations as below,

$$ \begin{aligned} &p_{1}^{a_{1}} M_{1}+\cdots+p_{m}^{a_{1}} M_{m}=M \bigl(p^{a_{1}}\bigr), \\ &\vdots \\ &p_{1}^{a_{m}} M_{1}+\cdots+p_{m}^{a_{m}} M_{m}=M\bigl(p^{a_{m}}\bigr). \end{aligned} $$

The equations above can be re-written as

$$ (P_{m} \otimes I_{n}) \begin{pmatrix} M_{1}\\ \vdots\\ M_{m} \end{pmatrix} = \begin{pmatrix} M(p^{a_{1}})\\ \vdots\\ M(p^{a_{m}}) \end{pmatrix}, $$

where \(I_{n} \in\mathbb {R}^{n\times n}\) is the identity matrix, and

$$ P_{m}= \begin{pmatrix} p_{1}^{a_{1}} &\ldots& p_{m}^{a_{1}}\\ \vdots& & \vdots\\ p_{1}^{a_{m}}& \ldots&p_{m}^{a_{m}} \end{pmatrix} \in\mathbb {R}^{m \times m}. $$

If \(p_{1}, p_{2}, \ldots, p_{m}\) are independent parameters, it is possible to select the samples \(p^{a_{i}}\) such that the corresponding \(m\times m\) matrix \(P_{m}\) in (7) is nonsingular, since otherwise, one or more parameters can be removed. Under this assumption,

$$ \begin{pmatrix} M_{1}\\ \vdots\\ M_{m} \end{pmatrix} = \bigl(P_{m}^{-1}\otimes I_{n} \bigr) \begin{pmatrix} M(p^{a_{1}})\\ \vdots\\ M(p^{a_{m}}) \end{pmatrix}, $$

where we use the following property of the Kronecker product: \((U \otimes Q)^{-1}=U^{-1}\otimes Q^{-1}\) for any nonsingular matrices \(U \in \mathbb {R}^{n_{U}\times n_{U}}\) and \(Q \in\mathbb {R}^{n_{Q}\times n_{Q}}\) [5]. Finally, the matrices \(M_{i}\) (\(i=1,\ldots, m\)) can be easily computed by

$$ \begin{aligned} &M_{1}=\tilde{p}_{11} M\bigl(p^{a_{1}}\bigr)+\cdots+\tilde{p}_{1m} M \bigl(p^{a_{m}}\bigr), \\ &\vdots \\ &M_{m}=\tilde{p}_{m1} M\bigl(p^{a_{1}}\bigr)+\cdots+ \tilde{p}_{mm} M\bigl(p^{a_{m}}\bigr), \end{aligned} $$

where \(\tilde{p}_{ij}\) is the \((i,j)\)-th entry of the matrix \(P_{m}^{-1}\):

$$ P_{m}^{-1}= \begin{pmatrix} \tilde{p}_{11} &\ldots& \tilde{p}_{1m}\\ \vdots& & \vdots\\ \tilde{p}_{m1} & \ldots&\tilde{p}_{mm} \end{pmatrix}. $$

An important property of the computation above is that it is independent of the large dimension n (typically several thousands or even higher) in linear system solves. To compute all \(M_{i}\) (\(M_{i}=E_{i}, A_{i}, B_{i}, C_{i}, D_{i}\), \(i=1,\ldots,m\)) in (3) for any large-scale matrices in (2), we need only to invert a small-scale \(m\times m\) matrix \(P_{m}\) once (m is typically below ten and equals 2 or 3 in our numerical examples) and conduct scalar-matrix multiplication and matrix addition with the dimension n, all of which are computationally much more efficient than solving (6) with its order mn. Note that the full inverse for such a small matrix can readily be computed, and multiplication with it is somewhat faster than working with the triangular factor and backward/forward solves due to the memory access pattern.

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

Various PMOR methods have been proposed in the literature, among which the methods based on multi-moment-matching are probably the easiest to implement and the most computationally efficient for many applications, especially for linear systems [2]. The multi-moment-matching PMOR method computes a basis matrix V based on the series expansion of the state vector x in the frequency domain. Under the zero initial condition, the frequency domain description for system (4) is

$$\begin{aligned} &\bigl(sE(p)-A(p)\bigr)x(s,p)=B(p)u(s), \\ &y(s,p)=C(p)x(s,p)+D(p)u(s), \end{aligned}$$

where we assume that the matrix pencil \((A(p),E(p))\) is regular for any p value, i.e., there exists \(\lambda_{p,0}\) such that \(\lambda _{p,0}E(p)-A(p)\) is nonsingular. Given expansion points \({p}^{0}=[p_{1}^{0},\ldots,p_{m}^{0}]\), and \(s_{0}\), \(x(s,p)\) in (10) can be expanded as

$$\begin{aligned} x(s,p)&=\bigl[I-(\sigma_{1}G_{1}+\cdots+ \sigma_{m}G_{m}+\sigma_{m+1}G_{m+1}+\cdots + \sigma_{2m}G_{2m})\bigr]^{-1}B_{M}u(s) \\ &=\sum_{i=0}^{\infty}(\sigma_{1}G_{1}+ \cdots+\sigma_{2m}G_{2m})^{i}B_{M}u(s), \end{aligned}$$

where \(\sigma_{i}=sp_{i}-s_{0}p_{i}^{0}\), \(\sigma_{m+i}=p_{i}-p_{i}^{0}\), \(G_{i}=-[s_{0}E(p^{0})-A(p^{0})]^{-1}E_{i}\), \(G_{m+i}=[s_{0}E(p^{0})-A(p^{0})]^{-1}A_{i}\), \(i=1,2,\ldots,m\), and \(B_{M}=[s_{0}E(p^{0})-A(p^{0})]^{-1} B(p)\), under the condition that all matrices in \([\cdot]^{-1}\) are nonsingular and

$$ \|\sigma_{1} G_{1} + \cdots+ \sigma_{2m} G_{2m} \| < 1. $$

Because of condition (12), the resulting ROM is normally accurate only around the expansion point \({p}^{0}\). To obtain a parametric ROM valid on a wider range, multiple expansion points are often employed as we will show below.


$$R_{j}=[G_{1},\ldots,G_{p}]R_{j-1}, \quad j=1,\ldots,q, $$


$$R_{0}=\bigl[s_{0}E\bigl(p^{0}\bigr)-A \bigl(p^{0}\bigr)\bigr]^{-1} [B_{1}, \ldots,B_{m}], $$

we can compute the matrix \(V_{s_{0}, p^{0},q}\), whose columns form an orthonormal basis of the subspace spanned by the first q of \(R_{i}\)’s:

$$ \operatorname{range}\{V_{s_{0}, p^{0},q}\}=\operatorname{span} \{R_{0}, R_{1},\ldots, R_{q} \}_{s_{0}, p^{0}}. $$

Using \(V:=V_{s_{0}, p^{0},q}\), which is assumed to be an \(n\times n_{r}\) matrix, we obtain the parametric reduced-order model via Galerkin projection,

$$ \begin{aligned} & \bigl(V^{T}E(p)V \bigr) \frac{dx_{r}(t,p)}{dt}= \bigl(V^{T}A(p)V \bigr)x_{r}(t,p)+ \bigl(V^{T}B(p) \bigr)u(t), \\ &y_{r}(t,p)= \bigl(C(p)V \bigr)x_{r}(t,p)+D(p)u(t), \end{aligned} $$

where the state vector \(x_{r}(t,p)\) is of order \(n_{r}\). When \(A(p), B(p), C(p), E(p)\) all take the affine form (3), the reduced parametric matrices can be computed by the formula

$$\begin{aligned} &V^{T}E(p)V=p_{1} V^{T} E_{1} V + \cdots+ p_{m} V^{T} E_{m} V, \\ &V^{T}A(p)V=p_{1} V^{T} A_{1} V + \cdots+ p_{m} V^{T} A_{m} V, \\ &V^{T}B(p)=p_{1} V^{T} B_{1} + \cdots+ p_{m} V^{T} B_{m}, \\ &C(p)V=p_{1} C_{1} V + \cdots+ p_{m} C_{m} V, \end{aligned}$$

where all constant matrices on the right-hand side can be pre-computed.

Note that the number of columns in \(R_{j}\) increases exponentially with j. When the number of the parameters in p is larger than 2, or when there are many inputs, multiple expansion points should be used to keep the size of the reduced-order model reasonable. The idea is straightforward. Given a set of expansion points \(s_{i}, p^{i}, i=0,\ldots, k\) (the superscript i for p is not a power: it only indicates the i-th expansion point), a matrix \(V_{s_{i},p^{i}}\) can be computed for each pair \((s_{i}, p^{i})\) as

$$ \operatorname{range}\{V_{s_{i},p^{i},q_{r}}\}=\operatorname{span} \{R_{0}, R_{1},\ldots, R_{q_{r}} \}_{s_{i},p^{i}}. $$

The final projection matrix V is obtained from the orthogonalization of all matrices \(V_{s_{i},p^{i},q_{r}}\),

$$ V= \operatorname{orth}\{V_{s_{0},p^{0},q_{r}}, \ldots, V_{s_{k},p^{k},q_{r}} \}. $$

For similar accuracy, the number \(q_{r}\) in (15) can usually be taken much smaller than q in (13) and normally, only a few well-chosen expansion points suffice. For example, 1 or 2 commonly suffices for \(q_{r}\), while q must be taken a much larger value depending on the problem. The reason is that, using multiple expansion points, the difficulty of the parametric dependence can be tackled by adding new interpolation points, each of which adds only a few columns due to the small \(q_{r}\), which is much more economical than using a single expansion point, where this difficulty must be treated with the increase of q, each step of which becomes increasingly more expensive. Consequently, the reduced-order model is normally smaller and more accurate on a broader parameter range when multiple expansion points are used.

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

For the general system (4) with \(l_{\mathrm{I}}\) inputs and \(l_{\mathrm{O}}\) outputs, the error bound \(\Delta(s,p)\) is defined as

$$\Delta(s,p)= \mathop{\max_{1\leq i\leq l_{\mathrm{I}},}}_{1\leq j \leq l_{\mathrm{O}}} \Delta_{ij}(s,p), $$

where \(\Delta_{ij}(s,p)\) is the error bound for the \((i,j)\)-th entry of the transfer function matrix of the ROM, i.e.,

$$\bigl|H_{ij}(s,p)-\hat{H}_{ij}(s,p)\bigr| \leq\Delta_{ij}(s,p), $$

where \(H(s,p)\) and \(\hat{H}(s,p)\) represent the transfer functions of the full-order model and the reduced-order model, respectively. In this paper, we define the \(\Delta_{ij}(s,p)\) as in [3], which is inspired by the a posteriori error bounds proposed for the reduced basis method [6]:

$$\Delta_{ij}(s,p)=\frac{\|r_{i}^{du}(s,p)\|_{2}\|r_{j}^{pr}(s,p)\|_{2}}{\beta (s,p)}+\bigl|\bigl(\hat{x}^{du} \bigr)^{*}r_{j}^{pr}(s,p)\bigr|, $$


$$\begin{aligned}& r_{j}^{pr}(s,p)=B(:,j)-[sE-A]\hat{x}^{pr}_{j}, \\& \hat{x}^{pr}_{j}=V\bigl(sV^{T}EV-V^{T}AV \bigr)^{-1}V^{T}B(:,j), \\& r_{i}^{du}(s,p)=-C(i,:)^{T}-\bigl[\bar{s}E^{T}-A^{T}\bigr]\hat{x}^{du}_{i}, \end{aligned}$$

is the conjugate of s, and the state \(x^{du}_{i}\) of the dual system is approximated by

$$\hat{x}^{du}_{i}=-V^{du}\bigl(\bar{s} \bigl(V^{du}\bigr)^{T}E^{T}V^{du}- \bigl(V^{du}\bigr)^{T}A^{T}V^{du} \bigr)^{-1}\bigl(V^{du}\bigr)^{T}C(i,:)^{T}. $$

Here, for ease of notation, p is dropped from the matrices \(E(p), A(p), B(p)\) and \(C(p)\), and the j-th column of \(B(p)\) and the i-th row of \(C(p)\) are denoted by \(B(:,j)\) and \(C(i,:)\), respectively. The variable \(\beta(s,p)\) is the smallest singular value of the matrix \(sE(p)-A(p)\). The matrix \(V^{du}\) can be computed, for example, using (15) and (16), but replacing \(R_{0}, \ldots, R_{q_{r}}\) with \(R_{0}^{du}, R_{1}^{du},\ldots, R_{q_{r}}^{du}\), where the matrices \(s_{i}E(p^{i})-A(p^{i})\) in \(R_{0}, \ldots, R_{q_{r}}\) are substituted by \(\bar{s}_{i} E^{T}(p^{i})-A^{T}(p^{i})\), and \(E_{j}\) by \(E^{T}_{j}\), \(A_{j}\) by \(A^{T}_{j}\), \(B_{j}\) by \(C(j,:)^{T}\), \(j=1,\ldots,m\). The derivation of \(\Delta(s,p)\) is detailed in [3].

Thanks to the error bound \(\Delta(s,p)\) for the ROM, the expansion points \((s_{i},p^{i})\) can be selected adaptively, and the projection matrix V can be computed automatically as is shown in Algorithm 1. It is worth pointing out that although the error bound is parameter-dependent, many p-independent terms constituting the error bound need to be pre-computed only once, which can be repeatedly used in the algorithm for all samples of p in \(\Xi_{\mathrm{train}}\), e.g., the terms \(V^{T}M_{1}V, \ldots, V^{T}M_{m} V\), etc.

Algorithm 1
figure a

Adaptively selecting expansion points ŝ, , and computing V automatically

5 Test models

To test the techniques proposed, we will use two applications arising from thermal and electro-thermal simulations: a package shown in Figure 1(a), whose purpose is to allow easy handling and assembly onto printed circuit boards and to protect the devices from damage [7], and a Power-MOS device shown in Figure 1(b), which is commonly used in energy harvesting, where energy from external sources like light and environmental heat is collected in order to power small devices such as implanted biosensors [8, 9]. With the scaling down of integrated circuits, thermal issues have attracted increasingly more attentions and become a major consideration in the design of integrated circuits.

Figure 1
figure 1

Physical models considered in the numerical tests.

The dynamics of both applications can be described by the same governing equations. The electrical sub-system can be described by

$$\begin{aligned} & \nabla\cdot J + \frac{\partial\rho}{\partial t} =0, \\ &J=\sigma\cdot E, \qquad E=-\nabla U, \\ &\rho=-\nabla\cdot(\epsilon\nabla U), \end{aligned}$$

where J is the current density, E is the electrical field, U is the electrical potential, σ is the electrical conductivity, ϵ is the permittivity, and ρ is the charge density. In this paper, we ignore both the local charging, i.e., \(\epsilon=0\) and \(\rho=0\), and the dependence of the electrical conductivity σ on temperature, i.e., the electrical sub-system is independent of the thermal sub-system, and obtain the following simplified governing differential equation, which is time-independent:

$$\nabla\cdot(\sigma\cdot\nabla U)=0. $$

The thermal sub-system is governed by similar equations:

$$\begin{aligned} &\nabla\cdot\phi_{q} + \frac{\partial w(T)}{\partial t} = Q, \\ &\phi_{q}=-\kappa\nabla T, \\ &w(T)=C_{T}(T-T_{\mathrm{ref}}), \end{aligned}$$

where \(\phi_{q}\) is the heat flux, w is the local energy storage, \(C_{T}\) is the thermal capacitance, and Q represents heat sources or sinks. For the thermal sub-system, we also ignore the dependence of the thermal capacitance on the temperature. For Q, we use two options in this paper.

• 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).

• The electro-thermal option takes Q as a coupling term from the electrical sub-system: the Joule self-heating that is of great importance in power-aware design of integrated circuits:

$$Q=Q_{\mathrm{SH}}=E\cdot J. $$

In this case, the whole system is one-way coupled: the thermal sub-system depends on the electrical sub-system, while the electrical sub-system is independent of the thermal system. The state space representation of the electro-thermal option is:

figure c

where the input vector \(u\in\mathbb{R}^{l_{\mathrm{I}}}\) represents the input voltages and temperatures at the contacts, the output vector \(y\in\mathbb{R}^{l_{\mathrm{O}}}\) represents the output voltages, currents, temperature, and thermal fluxes at the contacts, \(A_{\mathrm{E}}(p)\in\mathbb{R}^{n_{\mathrm{E}}\times n_{\mathrm{E}}}\), \(B_{\mathrm{E}}(p)\in\mathbb{R}^{n_{\mathrm{E}}\times l_{\mathrm{I}}}\), \(E_{\mathrm{T}}(p)\in\mathbb{R}^{n_{\mathrm{T}}\times n_{\mathrm{T}}}\), \(A_{\mathrm{T}}(p)\in\mathbb{R}^{n_{\mathrm{T}}\times n_{\mathrm{T}}}\), \(B_{\mathrm{T}}(p)\in\mathbb{R}^{n_{\mathrm{T}}\times l_{\mathrm{I}}}\), \(C_{\mathrm{E}}(p)\in\mathbb{R}^{l_{\mathrm{O}}\times n_{\mathrm{E}}}\), \(C_{\mathrm{T}}(p)\in\mathbb{R}^{l_{\mathrm{O}}\times n_{\mathrm{T}}}\), \(D_{\mathrm{T}}(p)\in\mathbb{R}^{l_{\mathrm{O}}\times l_{\mathrm{I}}}\), and the tensor \(F(p)\in\mathbb{R}^{n_{\mathrm{T}}\times n_{\mathrm{E}} \times n_{\mathrm{E}}}\), which can be considered as \(n_{\mathrm{T}}\) slices of \(n_{\mathrm{E}}\) by \(n_{\mathrm {E}}\) matrices \(F_{i}(p) \in\mathbb{R}^{n_{\mathrm{E}} \times n_{\mathrm{E}}}\), \(i=1, \ldots, n_{\mathrm{T}}\), represents the nonlinear coupling of the electrical part with the thermal part. Denoting the i-mode tensor-matrix product by \(\times_{i}\) [10], the product \(F(p) \times_{2} x_{\mathrm{E}}(p)\times_{3} x_{\mathrm{E}}(p)\) is a vector of length \(n_{\mathrm{T}}\), whose i-th component is the standard vector-matrix-vector product \(x_{\mathrm{E}}(p)^{T} F_{i}(p) x_{\mathrm{E}}(p)\). In this formulation, the algebraic equation (17a) describes the electrical part, the ordinary differential equation (17b) describes the thermal part, in which the tensor \(F(p)\) describes Joule self-heating, (17c) specifies the initial conditions, and (17d) computes the output obtained from the electrical and thermal state vectors. Theoretically, Joule self-heating should be modeled by two tensor products: \(F(p) \times_{2} x_{\mathrm{E}}(p) \times_{3} x_{\mathrm{E}}(p)\) and \(G(p) \times_{2} x_{\mathrm{E}}(p) \times_{3} u(t)\). However, the influence of the second part is rather limited, and is therefore ignored in this paper. Instead of a single coupled system, we write out the electrical and thermal sub-systems explicitly to show the one-way coupling. Furthermore, our numerical results proved that PMOR is computationally much more efficient if we apply it to the algebraic equations and the ordinary differential equations separately rather than apply it to a single set of differential algebraic equations.

6 Parametric modeling and PMOR for the test models

For the package model shown in Figure 1(a), the parameter p is chosen to be the top layer thickness of the package, namely h. The finite-integration technique (FIT) for the modeling of the package leads to thermal fluxes that are proportional to the dual areas of the mesh cells and inversely proportional to the lengths of the edges in the mesh cells. Therefore, when considering meshes that are topologically equivalent for different package thicknesses, the parametric dependence of the matrices will take the form

$$M(h)=M_{0}+hM_{1}+\frac{1}{h}M_{2}\quad (M=A_{\mathrm{E}}, B_{\mathrm{E}}, E_{\mathrm{T}}, A_{\mathrm{T}}, B_{\mathrm{T}}, F, C_{\mathrm{E}}, C_{\mathrm{T}}, D). $$

The second term originates from the linear dependence of dual areas corresponding to the cell edges perpendicular to the thickness orientation, whereas the third term originates from dual areas associated to cell edges tangential to the thickness orientation [11]. It is clear that the above formulation is a special case of (3) with \(p_{1}=1\), \(p_{2}=h\), and \(p_{3}=\frac{1}{h}\).

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.

For the Power-MOS circuit model shown in Figure 1(b), the conductivity of the third metal layer, which we denote by σ, is chosen to be the parameter. The finite-integration technique (FIT) assembles fluxes that are proportional to the conductivity of each mesh cell material, and therefore, the parametric dependence of the matrices will take the form

$$M(\sigma)=M_{0}+\sigma M_{1}\quad (M=A_{\mathrm{E}}, B_{\mathrm{E}}, E_{\mathrm{T}}, A_{\mathrm{T}}, B_{\mathrm{T}}, F, C_{\mathrm {E}}, C_{\mathrm{T}}, D). $$

It is clearly in the form (3) by assigning \(p_{1}=1\) and \(p_{2}=\sigma\), and the proposed procedure for extracting the matrices/tensors can readily be used.

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.

• If we ignore the nonlinear coupling term in the thermal sub-system (17b), it is already in the form (4). To use the methods developed, we first conduct the Laplace transform to obtain its frequency domain representation

$$ \bigl( A_{\mathrm{T1}} + \sigma A_{\mathrm{T2}} - sE_{\mathrm{T1}} - (\sigma s) E_{\mathrm{T2}} \bigr)X = \bigl[B_{\mathrm{T1}} B_{\mathrm{T2}} A_{\mathrm{T1}} {x}_{\mathrm{T}}^{0} A_{\mathrm{T2}} {x}_{\mathrm{T}}^{0} \bigr] \left [ \textstyle\begin{array}[c]{c} -U \\ -\sigma U \\ \frac{-1}{s}\\ \frac{-\sigma}{s} \end{array}\displaystyle \right ]. $$

Then, we apply Algorithm 1 to system (18) to obtain the basis for the thermal sub-system, which we denote by \(V_{\mathrm{T}}\).

• To obtain a ROM for (17b), we approximate \(x_{\mathrm{E}}\) by \(V_{\mathrm{E}}\widehat {x}_{\mathrm{E}}\) and \(x_{\mathrm{T}}\) by \(V_{\mathrm{T}}\widehat {x}_{\mathrm{T}}\), and then force the approximation error to be orthogonal to the range of \(V_{\mathrm{T}}\). The resulting parametric ROM is

$$ \widehat{E}_{\mathrm{T}}(p) \dot{\widehat{x}}_{\mathrm{T}}(t,p) = \widehat{A}_{\mathrm{T}}(p) \widehat{x}_{\mathrm{T}}(t,p) + \widehat {B}_{\mathrm{T}}(p) u + \widehat{F}(p) \times_{2} \widehat {x}_{\mathrm{E}}(p) \times_{3} \widehat{x}_{\mathrm{E}}(p), $$

where \(\widehat{E}_{\mathrm{T}}(p)=V_{\mathrm{T}}^{T} E_{\mathrm{T}}(p) V_{\mathrm{T}}\), \(\widehat{A}_{\mathrm{T}}(p)=V_{\mathrm{T}}^{T} A_{\mathrm{T}}(p) V_{\mathrm{T}}\), \(\widehat{B}_{\mathrm{T}}(p)=V_{\mathrm{T}}^{T} B_{\mathrm{T}}(p)\), \(\widehat{F}(p)=F(p) \times_{1} V_{\mathrm{T}} \times_{2} V_{\mathrm{E}} \times_{3} V_{\mathrm{E}}\). To obtain the reduced tensor \(\widehat{F}(p)\), we first approximate \(x_{\mathrm{E}}(p)\) in the range of \(V_{\mathrm{E}}\), and then project the approximation onto the test subspace \(V_{\mathrm{T}}\), i.e., the tensor product \(\widehat{F}(p) \times_{2} \widehat {x}_{\mathrm{E}}(p) \times_{3} \widehat{x}_{\mathrm{E}}(p)\) equals \(V_{\mathrm{T}}^{T} [F(p) \times_{2} (V_{\mathrm{E}}\widehat {x}_{\mathrm{E}}(p)) \times_{3} (V_{\mathrm{E}}\widehat{x}_{\mathrm{E}}(p)) ]\). The advantage of the tensor formulation for the ROM is that using the reduced tensor, evaluating the ROM does not require computations with quantities of the order of the FOM. In our actual computations, the parametric matrices in the ROM are computed by

$$ \widehat{Y}(p)=\widehat{Y}_{\mathrm{c}}+p \widehat{Y}_{\mathrm{v}},\quad \widehat{Y} \in\{ \widehat{A}_{\mathrm{T}}, \widehat{B}_{\mathrm{T}}, \widehat{C}_{\mathrm{T}}, \widehat{E}_{\mathrm{T}}, \widehat {F} \}, $$

where \(\widehat{Y}_{\mathrm{c}}\) and \(\widehat{Y}_{\mathrm{v}}\) are pre-computed during the construction of the ROM. This precomputation is also applied to the electrical sub-system (17a) and the output computation (17d).

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

The package model is a multi-input multi-output system, with 34 inputs and 68 outputs. Algorithm 1 is employed to compute the parametric reduced-order model automatically. We used 6 samples of the package thickness \(h \in(0\ \mbox{$\upmu$m}, 100\ \mbox{$\upmu$m}]\), and one sample of \(s=2\pi f \jmath\), \(f \in[0\ \mbox{Hz}, 10^{8}\ \mbox{Hz}]\): \(s_{0}=200 \pi\jmath\), \(\jmath=\sqrt{-1}\), to constitute the training set \(\Xi_{\mathrm{train}}\) in Algorithm 1. The algorithm essentially selects the expansion points for p, since we fix s to the single expansion point \(s_{0}\). Only two iterations and two expansion points selected are required for convergence. The reduced-order model is of size \(r=58\). For each selected expansion point, we construct \(V_{ s_{i}, p^{i}}\) with only two terms \(R_{0}\) and \(R_{1}\) (\(q_{r}=1\) in Steps 6-7, Algorithm 1) in order to avoid the exponential increase in \(R_{j}\), \(j>1\). Table 1 lists the iterations and the error bounds at each iteration step.

Table 1 \(\pmb{V_{ s_{i}, p^{i}}=\operatorname{span}\{R_{0}, R_{1} \}_{ s_{i}, p^{i}}}\) , \(\pmb{i=1,2}\) , \(\pmb{\varepsilon_{\mathrm{tol}}=10^{-3}}\) , \(\pmb{n=8,549}\) , \(\pmb{r=58}\)

Figure 2(a) and (b) plot the temperature and the current at two different parts of the package, respectively. The temperature is of big magnitude, while the current is of very small magnitude, showing that there is no current at that part of the circuit. The reduced-order model catches the accuracy of both at 120 samples of p, and 100 time steps for each sample.

Figure 2
figure 2

Comparison between FOM-based and ROM-based simulation results of the package model, where both the order-8549 FOM and the order-58 ROM have 34 inputs and 68 outputs.

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.

The system is parameterized by the thickness of the top layer and excited by the inputs:

$$ u_{i}=\left \{ \textstyle\begin{array}{@{}l@{\quad}l} 1, & i=1, \\ 0, & 2\leq i \leq17, \\ 75\times10^{8} t + 75, &i=18, t\in[0, 10^{-8}],\\ 150 & i=18, t>10^{-8}, \\ 75 & 19 \leq i \leq34. \end{array}\displaystyle \right . $$

The initial condition for all electrical state variables is 0 V, and the initial condition for all thermal state variables is 75C. For the electrical sub-system, the training set is \(\{1, 2, 5, 8, 10, 20, 30, 40, 50, 60, 70, 80, 90, 100\}\) (in \(\ \mbox{$\upmu$m}\)), while for the thermal sub-system, the training set \(\{(s, h)\}\) contains 20 samples, in which the frequency (s) and the thickness of the top metal layer (h) are uniformly chosen within the ranges and \((0\ \mbox{$\upmu$m}, 30\ \mbox{$\upmu$m}]\), respectively. Using the PMOR method proposed, the electrical sub-system is reduced from order 1,122 to order 68, the thermal system is reduced from order 8,071 to order 606, and the speedup factor for the electro-thermal simulation is 7.2. The convergence behavior of the adaptive PMOR method is shown in Table 2 and the thermal flux output \(y_{36}\) and its relative error are shown in Figure 3.

Figure 3
figure 3

The thermal flux output \(\pmb{y_{36}}\) and its relative error for the package model.

Table 2 Convergence behavior of electro-thermal simulation of the package model ( \(\pmb{\epsilon_{\mathrm{tol}}=10^{-4}}\) )

Then, we apply the proposed methods to the Power-MOS circuit model, which has 6 inputs and 12 outputs. The system is parameterized by the conductivity of the third metal layer and excited by the inputs:

$$ u_{i}=\left \{ \textstyle\begin{array}{@{}l@{\quad}l} 0, & i=1,2, \\ 10^{7} t, &i=3, t\in[0, 10^{-6}],\\ 10, & i=3, t>10^{-6}, \\ 26.85, & i=4,5,6. \end{array}\displaystyle \right . $$

The initial condition for all electrical state variables is 0 V, and the initial condition for all thermal state variables is 26.85 C. For the electrical sub-system, the training set is , while for the thermal sub-system, the training set \(\{(s, \sigma)\}\) contains 20 samples, in which the frequency (s) and the conductivity of the top metal layer (σ) are uniformly chosen within the ranges and , respectively. Using the PMOR methods proposed, the electrical sub-system is reduced from order 1,160 to order 2, the thermal sub-system is reduced from order 11,556 to order 35, and the speedup factor for the electro-thermal simulation is 65.93. The convergence behavior of the adaptive PMOR method is shown in Table 3 and the thermal flux output \(y_{7}\) along with its relative error is shown Figure 4.

Figure 4
figure 4

The thermal flux output \(\pmb{y_{7}}\) and its relative error for the Power-MOS model.

Table 3 Convergence behavior of electro-thermal simulation of the Power-MOS model ( \(\pmb{\epsilon_{\mathrm{tol}}=10^{-12}}\) )

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.


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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.

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Correspondence to Yao Yue.

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The main idea of the paper was proposed by LF. The numerical simulations were mainly done by LF, YY and NB. The manuscript was initially prepared by YY. PB did thorough correction of the manunscript. Authors from MAGWEL provided the data (models, matrices) for numerical tests. All authors read and approved the final manuscript.

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Feng, L., Yue, Y., Banagaaya, N. et al. Parametric modeling and model order reduction for (electro-)thermal analysis of nanoelectronic structures. J.Math.Industry 6, 10 (2016).

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