Reduced-order modeling of nonlinear structural dynamical systems via element-wise stiffness evaluation procedure combined with hyper-reduction

Abstract

In nonlinear analysis, performing iterative inverse calculation and nonlinear system construction procedures incurs expensive computational costs. This paper presents an element-wise stiffness evaluation procedure combined with hyper-reduction reduced-order modeling (HE-STEP ROM) method. The proposed approach constructs a non-intrusive reduced-order model based on an element-wise stiffness evaluation procedure (E-STEP) and hyper-reduction methods. Because the E-STEP evaluates nonlinear stiffness coefficients element-by-element using cubic polynomial, numerous number of polynomial variables are required. The number of variables directly affects the computational efficiency of the online and offline stages. Therefore, to enhance efficiency of the online/offline stages, the proposed method employs hyper-reduction method. By applying hyper-reduction, the full stiffness coefficients are approximated from the stiffness coefficients evaluated at a few sampling points. Subsequently, the number of polynomial equations and variables is prominently reduced, and the efficiency of the reduced system increases. The efficiency and accuracy of the proposed approach are validated via several structural dynamic problems with geometric and material nonlinearities.

Appendices

Appendix A : Stiffness evaluation procedure using the least squares method

To clarify the explanation, the displacement combination and stiffness coefficient matrices in Equation (2) are newly defined, as follows:

$$ \Gamma _{i} \left( {\mathbf{u}} \right) = \Gamma _{i} \left( {\mathbf{u}} \right)_{1} + \Gamma _{i} \left( {\mathbf{u}} \right)_{2} \cdots + \Gamma _{i} \left( {\mathbf{u}} \right)_{En} \approx \mathop \sum \limits_{m = 1}^{En} \left[ {K_{ijkl}^{{}} u_{j} u_{k} u_{l} } \right]_{m} $$

(A1)

where K_ijkl includes $K_{ij}^{1} ,K_{ijk}^{2}$, and $K_{ijkl}^{3}$ in Equation (A1). u_ju_ku_l is the displacement combination, which includes 1 as its value, and can also be expressed as u_j and u_ju_k ($u_{j} \cdot 1 \cdot 1$ and $u_{j} u_{k} \cdot 1$).

If the sample data sets, with internal force and displacement, are collected from the sampling analysis, the internal force model of Equation (A1) for the mth element can be represented as follows:

$$ \left[ {\begin{array}{*{20}c} {\Gamma _{11} } & \cdots & {\Gamma _{1S} } \\ \vdots & \ddots & \vdots \\ {\Gamma _{n1} } & \cdots & {\Gamma _{nS} } \\ \end{array} } \right]_{m} = \left[ {\begin{array}{*{20}c} {\tilde{K}_{11} } & \cdots & {\tilde{K}_{1p} } \\ \vdots & \ddots & \vdots \\ {\tilde{K}_{n1} } & \cdots & {\tilde{K}_{np} } \\ \end{array} } \right]_{m} \times \left[ {\begin{array}{*{20}c} {\tilde{u}_{11} } & \cdots & {\tilde{u}_{1S} } \\ \vdots & \ddots & \vdots \\ {\tilde{u}_{p1} } & \cdots & {\tilde{u}_{pS} } \\ \end{array} } \right]_{m} $$

(A2)

where m, n, p, and S are the element number, number of degrees of freedom for one element, number of displacement combinations for one element, and quantity of sample data, respectively. $\tilde{K}$ and $\tilde{u}$ indicate one combination of K_ijkl and u_ju_ku_l in Equation (A1), respectively. Γ_m and $\tilde{\varvec{u}}_{m}$ are the sampled internal force and displacement combination matrix in Equation (A2), respectively. Equation (A2) can be expressed as ${\varvec{\Gamma}}_{m} = \tilde{\varvec{K}}_{m} \tilde{\varvec{u}}_{m}$. The least squares method is used to evaluate the stiffness coefficient matrix, $\tilde{\varvec{K}}_{m}$, even if it is not a square matrix. Thus, the stiffness coefficient matrix can be solved for displacement combinations using the least squares method, as follows:

$$ \left[ {\begin{array}{*{20}c} {\Gamma _{11} } & \cdots & {\Gamma _{1S} } \\ \vdots & \ddots & \vdots \\ {\Gamma _{n1} } & \cdots & {\Gamma _{nS} } \\ \end{array} } \right]_{m} \left[ {\begin{array}{*{20}c} {\tilde{u}_{11} } & \cdots & {\tilde{u}_{1S} } \\ \vdots & \ddots & \vdots \\ {\tilde{u}_{p1} } & \cdots & {\tilde{u}_{pS} } \\ \end{array} } \right]_{m}^{ - 1} = \left[ {\begin{array}{*{20}c} {\tilde{K}_{11} } & \cdots & {\tilde{K}_{1p} } \\ \vdots & \ddots & \vdots \\ {\tilde{K}_{n1} } & \cdots & {\tilde{K}_{np} } \\ \end{array} } \right]_{m} $$

(A3)

The above process is performed for all elements and a stiffness coefficient suitable for each degree of freedom in the global system is assembled.

Appendix B: GappyPOD + E deterministic sampling strategy

The L₂ error of the approximation model satisfies the following equation:

$$ \|{\varvec{\Gamma}}^{nl} - {\tilde{\mathbf{\Gamma }}}^{nl}\|_{2} \le \|\left( {{\mathbf{Z}}^{\text{T}} {\varvec{\Omega}}} \right)^{\dag }\|_{2} \|{\varvec{\Gamma}}^{nl} - {\mathbf{\Omega \Omega }}^{\text{T}} {\varvec{\Gamma}}^{nl}\|_{2} $$

(B1)

Here, $\|{\varvec{\Gamma}}^{nl} - {\mathbf{\Omega \Omega }}^{\text{T}} {\varvec{\Gamma}}^{nl}\|_{2}$ affects the approximation quality of the subspace spanned by Ω, and $\|\left( {{\mathbf{Z}}^{\text{T}} {\varvec{\Omega}}} \right)^{\dag }\|_{2}$ quantifies the effectiveness of the sampling points.

To achieve robustness and minimize the quantity $\|\left( {{\mathbf{Z}}^{\text{T}} {\varvec{\Omega}}} \right)^{\dag }\|_{2}$ with a small number of sampling points, Refs. [40] proposed the GappyPOD + E (where E is the eigenvector) deterministic sampling strategy. The GappyPOD + E algorithm is based on the lower bound for the smallest eigenvalues of the updated matrix, which was introduced in [41]. The minimization problem is equal to maximizing the smallest eigenvalue of ${\mathbf{Z}}^{\text{T}} {\varvec{\Omega}}$.

$$ \|\left( {{\mathbf{Z}}^{\text{T}} {\varvec{\Omega}}} \right)^{\dag }\|_{2} = s_{max} \left( {\left( {{\mathbf{Z}}^{\text{T}} {\varvec{\Omega}}} \right)^{\dag } } \right) = \frac{1}{{s_{min} \left( {{\mathbf{Z}}^{\text{T}} {\varvec{\Omega}}} \right)}} $$

(B2)

where $s_{max} \left( {\mathbf{B}} \right)$ and $s_{min} \left( {\mathbf{B}} \right)$ indicate the largest and smallest singular values of B, respectively. The GappyPOD + E algorithm follows the lower bounds of the smallest eigenvalues to select the sampling points that maximize $s_{min} \left( {\mathbf{B}} \right)$; this is achieved by leveraging the eigenvector corresponding to the smallest eigenvalue. In this paper, the relevant summary is explained, and a detailed explanation and formulation can be found in Refs. [40]. If the authors add a sampling point, then ${\mathbf{Z}}_{p + 1}^{T} {\varvec{\Omega}} \in {\mathbb{R}}^{{\left( {p + 1} \right) \times N}}$ becomes

$$ {\mathbf{Z}}_{p + 1}^{T} {\varvec{\Omega}} = \left[ {\begin{array}{*{20}c} {{\mathbf{Z}}_{p}^{T} {\varvec{\Omega}} } \\ {\varvec{\omega}_{ + } } \\ \end{array} } \right], $$

(B3)

where $\varvec{\omega}_{ + } \in {\mathbb{R}}^{1 \times N}$ is the row of Ω for the new sampling point. Zimmermann et al. [46] demonstrated that the change in the singular values of ${\mathbf{Z}}_{p}^{T} {\varvec{\Omega}}$ to ${\mathbf{Z}}_{p + 1}^{T} {\varvec{\Omega}}$ can be computed using a symmetric rank-one update.

$$ {\mathbf{Z}}_{p + 1}^{T} {\varvec{\Omega}} = \left[ {\begin{array}{*{20}c} {\varvec{V}_{p} } & 0 \\ 0 & 1 \\ \end{array} } \right]\left[ {\begin{array}{*{20}c} {{\varvec{\Sigma}}_{p} } \\ {\varvec{\omega}_{ + } \varvec{W}_{p} } \\ \end{array} } \right]\varvec{W}_{p}^{T} , $$

(B4)

where ${\mathbf{Z}}_{p}^{T} {\varvec{\Omega}} = \varvec{V}_{p} {\varvec{\Sigma}}_{p} \varvec{W}_{p}^{T}$ is computed using the SVD method. The singular values of ${\mathbf{Z}}_{p + 1}^{T} {\varvec{\Omega}}$ are calculated by the square roots of the eigenvalues of $\left( {{\mathbf{Z}}_{p + 1}^{T} {\varvec{\Omega}} } \right)^{T} {\mathbf{Z}}_{p + 1}^{T} {\varvec{\Omega}}$.

$$ \left( {{\mathbf{Z}}_{p + 1}^{T} {\varvec{\Omega}} } \right)^{T} {\mathbf{Z}}_{p + 1}^{T} {\varvec{\Omega}} = \varvec{W}_{p} \left( {{\varvec{\Sigma}}_{p}^{2} + \varvec{W}_{p}^{T}\varvec{\omega}_{ + }^{T}\varvec{\omega}_{ + } \varvec{W}_{p} } \right)\varvec{W}_{p}^{T} = \varvec{W}_{p} {\varvec{\Lambda}}_{p + 1} \varvec{W}_{p}^{T} , $$

(B5)

where the square roots of the eigenvalues of ${\varvec{\Lambda}}_{p + 1}$ are singular values of ${\mathbf{Z}}_{p + 1}^{T} {\varvec{\Omega}}$. Let $\uplambda_{1}^{{\left( {p + 1} \right)}} , \ldots ,\uplambda_{N}^{{\left( {p + 1} \right)}}$ be the eigenvalues of ${\varvec{\Lambda}}_{p + 1}$ and $\uplambda_{1}^{\left( p \right)} , \ldots ,\uplambda_{N}^{\left( p \right)}$ be the eigenvalues of ${\varvec{\Sigma}}_{p}^{2}$, both arranged in descending order. The GappyPOD + E algorithm aims to select a row Ω that maximizes the smallest eigenvalue $\uplambda_{N}^{{\left( {p + 1} \right)}}$. From Weyl’s theorem [47], we have $\uplambda_{N}^{{\left( {p + 1} \right)}} \ge \uplambda_{N}^{\left( p \right)}$, which shows that the additional sampling point will not increase the eigenvalues of $\|\left( {{\mathbf{Z}}_{p + 1}^{T} {\varvec{\Omega}}} \right)^{\dag }\|_{2}$, compared to the eigenvalues of $\|\left( {{\mathbf{Z}}_{p}^{T} {\varvec{\Omega}}} \right)^{\dag }\|_{2}$.

In Refs. [40], the lower bounds for the eigenvalues of the updated matrices, for selecting the sampling point to rapidly increase the smallest eigenvalue, are calculated using the following equation:

$$ {\text{g}} + \|\bar{\varvec{\omega }}_{ + }\|_{2}^{2} - \sqrt {\left( {{\text{g}} + \|\bar{\varvec{\omega }}_{ + }\|_{2}^{2} } \right)^{2} - 4{\text{g}}\left( {\varvec{z}_{N}^{\left( p \right)^{T}} \bar{\varvec{\omega }}_{ + } } \right)^{2} ,} $$

(B6)

where$\bar{\varvec{\omega }}_{ + } = \varvec{W}_{p}^{T}\varvec{\omega}_{ + }^{T}$. $g = \uplambda_{N - 1}^{\left( p \right)} - \uplambda_{N}^{\left( p \right)}$ indicates the eigengap and ${\varvec{\Sigma}}_{p}^{2}$ is the diagonal matrix, with diagonal elements sorted in descending order; thus, $\varvec{z}_{N}^{\left( p \right)} \in {\mathbb{R}}^{N}$ is the Nth canonical unit vector of dimension N. A detailed explanation and derivation of the bound equation can be found in Refs. [40].