Energy-based Method Research Articles

An Energy-Based Discontinuous Galerkin Method for the Nonlinear Schrödinger Equation with Wave Operator

An Energy-Based Discontinuous Galerkin Method for the Nonlinear Schrödinger Equation with Wave Operator

Robustness analysis of dynamic progressive collapse of precast concrete beam–column assemblies using dry connections under uniformly distributed load

A series of dynamic progressive collapse tests using the uniformly distributed load (UDL) was conducted, to analyze the robustness of three precast concrete (PC) beam–column assemblies using dry connections of top-and-seat angles (TSA-D), strengthened top-and-seat angles (STSA-D), and high-ductility reinforcement (DSTSA-D), under a middle column removal scenario. Finite element (FE) models were also developed to accurately simulate the collapse process. Based on the tests and FE models, comparative studies were conducted on dynamic and static collapse responses of the PC assemblies with identical configuration, loading regime, and boundary conditions. The results showed that: the horizontal reaction force developments were similar under dynamic and static collapse scenarios; under dynamic collapse scenarios, the initial stiffness of the PC assemblies increased due to the strain rate effect, while the peak vertical reaction forces under the compressive arch action (CAA) were smaller than those under static collapse scenarios, attributed to dynamic damage; under both dynamic and static collapse scenarios under UDL, the beam deformed in a downward convex curved shape, with local material deformation becoming more concentrated under dynamic collapse scenarios. An energy-based method for calculating dynamic collapse resistance was evaluated and revised: 1) the traditional method, which converts static responses from static analysis into dynamic responses was found to underestimate the resistance under small deformations due to neglecting the strain rate effect, and to overestimate ultimate resistance by ignoring dynamic damage; 2) by using dynamic vertical reaction forces instead, a more accurate prediction of the dynamic collapse resistance was achieved with a single dynamic collapse analysis. Additionally, dynamic amplification factors for the PC assemblies were calculated based on FE models and static test results providing basic knowledge in whole PC structural level research.

Energy-based damage assessment method for masonry walls under seismic and fire loads

The evaluation of the out-of-plane (OOP) behavior for infilled walls is a critical aspect of regional seismic and fire resilience assessment. This study presents a detailed finite element (FE) model that comprehensively captures the seismic and fire behaviors of masonry walls. The FE model incorporates key factors governing the thermo-mechanical behavior of masonry walls, including temperature-dependent material properties, nonlinearities in geometry and material, block cracking and crushing, and mortar damage. Then, an energy-based damage criterion is proposed based on the principle of energy conservation, leveraging the combined mixed fracture energy and internal energy of elements. In addition, this study defines the damage levels of masonry walls under OOP loading and fire and determines corresponding damage indices using the proposed energy-based damage criterion. This criterion is further compared with a stiffness-based damage criterion. Finally, the proposed damage assessment method is employed to evaluate the damage evolution of infilled walls under seismic, fire and post-earthquake fire scenarios, and to conduct a fragility analysis. The results demonstrate a high degree of agreement between the predicted and measured damage states of masonry walls under seismic and fire loads. More severe seismic damage does not always result in diminished fire resilience. Evaluating post-earthquake fire damage requires considering the impact of displacement directions caused by earthquakes and fires.

Topology optimization of extreme mechanical metamaterials considering the anisotropy of additive manufactured parts

Abstract Additive manufacturing (AM) has the advantage of fabricating complex geometries designed by topology optimization. However, the layer-by-layer stacking of AM causes the anisotropic property of the manufactured parts, which is rarely considered in the topology optimization of metamaterials. Hence, this paper presents a new method for the topology optimization of metamaterials with anisotropic properties. First, the elastic moduli in different directions of anisotropic materials are introduced into the solid isotropic material with penalization interpolation function. Then, the effective elastic properties of anisotropic metamaterials are evaluated using the energy-based homogenization method, and the formulation of the topology optimization of anisotropic metamaterials is constructed, and it is iteratively solved by the method of moving asymptotes algorithm. Finally, several two-dimensional and three-dimensional numerical examples and a set of mechanical experiment are conducted to demonstrate the effectiveness of the proposed method.

Evaluating dynamic models for rigid-foldable origami: unveiling intricate bistable dynamics of stacked-Miura-origami structures as a case study.

Recent advances in origami science and engineering have particularly focused on the challenges of dynamics. While research has primarily focused on statics and kinematics, the need for effective and processable dynamic models has become apparent. This paper evaluates various dynamic modelling techniques for rigid-foldable origami, particularly focusing on their ability to capture nonlinear dynamic behaviours. Two primary methods, the lumped mass-spring-damper approach and the energy-based method, are examined using a bistable stacked Miura-origami (SMO) structure as a case study. Through systematic dynamic experiments, we analyse the effectiveness of these models in predicting bistable dynamic responses, including intra- and interwell oscillations, in different loading conditions. Our findings reveal that the energy-based approach, which considers the structure's inertia and utilizes dynamic experimental data for parameter identification, outperforms other models in terms of validity and accuracy. This model effectively predicts the dynamic response types, the rich and complex nonlinear characteristics and the critical frequency where interwell oscillations occur. Despite its relatively increased complexity in model derivation, it maintains computational efficiency and shows promise for broader applications in origami dynamics. By comparing model predictions with experimental results, this study enhances our understanding of origami dynamics and contributes valuable insights for future research and applications. This article is part of the theme issue 'Origami/Kirigami-inspired structures: from fundamentals to applications'.

Should We Conserve Entropy or Energy when Computing CAPE with Mixed-Phase Precipitation Physics?

Abstract The rapidly increasing resolution of global atmospheric reanalysis and climate model datasets necessitates finding methods for computing convective available potential energy (CAPE) both efficiently and accurately. To this end, this article compares two common methods for computing CAPE which conserve either energy or entropy. Inaccuracies in these computations arise from both physical and numerical errors. For instance, computing CAPE with entropy conserved results in physical errors from nonequilibrium phase transitions but minimizes numerical errors because solutions are analytic at each height. In contrast, computing CAPE with energy conserved avoids these physical errors, but accumulates numerical errors that are grid-resolution-dependent because the numerical integration of a differential equation is required. Analysis of CAPE computed with large databases of soundings from the tropical Amazon and midlatitude storm environments shows that physical errors from the entropy method are typically 1%–3% as large as CAPE, which is comparable to the numerical errors from conserving energy with grid spacing of 25 and 250 m using explicit first-order and second-order integration schemes, respectively. Errors in entropy-based CAPE calculations are also insensitive to vertical grid spacing, in contrast to energy-based calculations whose error strongly scales with the grid spacing. It is shown that entropy-based methods are advantageous when intercomparing datasets with differing vertical resolution because they produce accurate and reasonably fast results that are insensitive to grid resolution, whereas a second-order energy-based method is advantageous when analyzing data with a consistent vertical resolution because of its superior computational efficiency. Significance Statement Convective available potential energy (CAPE) is a measure of instability in the atmosphere that helps forecasters and researchers understand when and where thunderstorms will form. The purpose of this article is to identify the most efficient and accurate methods for computing CAPE. Two methods are considered here, one that relates to the entropy (a measure of thermodynamic disorder) of an air parcel and one that relates to the energy of an air parcel. Results indicate that the entropy method is most accurate and insensitive to the resolution of the data used for the calculation (which can vary considerably), whereas the energy method uses the least computation time.

New Results on Rapid Dynamical Pattern Recognition via Deterministic Learning From Sampling Sequences.

Rapid dynamical pattern recognition based on the deterministic learning method (DLM-based RDPR) aims to rapidly recognize the most similar dynamical pattern pair from perspectives of differences in inherent system dynamics. The basic mechanism is to use available recognition errors to reflect the differences in the dynamics of dynamical pattern pairs and then to make a decision based on a minimal recognition error (MRE) principle. This article focuses on providing a rigorous theoretical analysis of the MRE principle in DLM-based RDPR under the sampled-data framework. Specifically, we seek a unified methodology from the similarity definition to the measure implementation and then to derive general sufficient conditions and necessary conditions for the MRE principle. The main idea is to: 1) from the average signal energy aspect, define a time-dependent dynamics-based similarity in dynamical pattern pairs and reestablish the measure of recognition errors generated from the DLM-based RDPR; 2) introduce the energy-based Lyapunov method to establish the interrelation between the dynamical distance and the recognition error; and 3) derive sufficient conditions and necessary conditions from two directions of the interrelation. The proposed conditions distinguish themselves from virtually all of the existing DLM-based RDPR works with only sufficient conditions in the sense that it is shown in a rigorous analysis that under what conditions, the pattern pair recognized based on the MRE principle is indeed the most similar one. Therefore, the proposed work makes the DLM-based RDPR possess good interpretability and provides strong theoretical guidance in engineering applications.

An analytical model for predicting residual stress in shot peening with strain energy method

In this study a model by novel analytical approach is developed and experimentally verified for shot peening residual stress distribution. The residual stress field induced by single shot impact is calculated by using Glinka–Molski energy-based method and kinematic hardening model. The formulation of the compressive residual stress (CRS) distribution is often based on plane strain or plane stress. It can be determined from the derived relation presented in this paper, the final residual stress in the full coverage conditions is the average of the two strain and stress plane expressions proposed by previous researchers. The distribution of residual stress is one of the key differences between the profiles produced by the results of the current model. There is a significant distinction between surface residual stress and maximum CRS, because the CRS profile near the surface is more curved compared to profiles obtained in earlier analytical models. The experimental data obtained by XRD analysis indicate the correctness and precision of the current model. Another goal of this study is to increase the fatigue life of GTD-450 stainless steel by shot peening at two different peening intensities. The fatigue life of samples were obtained by rotary bending test. Analytical results that confirmed by experimental findings shown bigger maximum compressive residual stresses occurred in higher shot peening intensities. This incident can improved fatigue life by deeper plastically deformed layer.

Mode selection and combination rules of energy-based multimodal pushover method for simply supported beam bridges

The energy-based method (EBM) has been improved as Energy-based Multimodal Pushover Method (EMPM), and is introduced and verified to be effective and convenient for evaluate the seismic performance of the simply supported beam bridges. This paper mainly focuses on the research of vibration mode selection and combination rules of EMPM. Firstly, the longitudinal vibration modes of simply supported beam bridge are categorized into five cases based on mode shapes, and the case definition is emphasized. Then, corresponding to each case, the Pushover loading force are established, and the modal energy, modal displacement, and modal energy capability curve are proposed. After that, the energy-based demand curve and the corresponding energy coefficient for each case is recommended. Finally, only the mode with maximum mode participation factor (MPF) of each case is suggested to take part in the calculation of the seismic response with the proposed mode participation factor combination rule. Case studies of a practice bridge under Chi-Chi and El Centro earthquake waves are used to illustrate the effectiveness and accuracy of the proposed mode selection and combination rules. The study found that the EMPM with the proposed rules could effectively and rapidly assessing the seismic performance of the simply supported beam bridge, resulting in a 70 % decrease in time consumption compared with the Nonlinear Time History Analysis Method (NTHAM). Compared with the CQC and SRSS combination results, the proposed Mode Participation Factor (MPF) combination rule reduced the maximum relative deviation from 39.4 % to 21.09 % between EMPM and NTHAM for peak displacement. Further research showed that the proposed mode selection approach only uses modes with the highest MPF in each case, decreased the number of modes involved more than 50 % and the result accuracy remained almost identical.

Strain rate- and temperature-dependent mechanical properties of Ti-6Al-4V in dynamic compression: hardening and softening behaviour analysis using strain energy-based method

Strain rate- and temperature-dependent mechanical properties of Ti-6Al-4V in dynamic compression: hardening and softening behaviour analysis using strain energy-based method

Research on Inter-Satellite Laser Ranging Scale Factor Estimation Methods for Next-Generation Gravity Satellites

The scale factor serves as a ruler for converting raw phase measurements into physical displacements and significantly impacts the preprocessing of data from the Laser Ranging Interferometer (LRI) in laser ranging systems. In the current GRACE Follow-On (GRACE-FO) mission for low–low tracking gravity satellites, most of the existing LRI scale factor estimation algorithms heavily rely on cross-calibration between instantaneous/biased ranges from the Ka-Band Ranging Interferometer (KBR) and the LRI. However, due to the nonlinearity of the objective function (which includes terms involving the product of scale and time shifts), the scale factor estimation may absorb errors from timing noise. Moreover, future gravity missions or gravity detection tasks may no longer incorporate KBR ranging instruments. To address these challenges, this paper proposes an energy-based method for scale factor estimation using inter-satellite baseline solutions. Comparative analysis indicates that the proposed method effectively disentangles two parameters in the objective function and can be applied in scenarios where KBR data are unavailable, demonstrating promising prospects for practical application. The experimental results show that when the KBR validation residuals are lower than 0.8 mm, the SYSU LRI1B V01 products exhibit residuals below the payload design accuracy thresholds in the frequency band of 2 mHz to 0.1 Hz. Additionally, the stability of the scale factors obtained from the baseline can reach 10−7. Although this is still below the required precision of better than 10−8 for the laser frequency stability in next-generation gravity satellites, advancements in orbit determination technology and the enhanced stability of GPS receivers offer potential for future precision improvements. Currently, this method appears suitable for roughly extracting the scale factor as a stochastic mean over several months or serving as a backup validation strategy for future missions, but it is not well suited to measure day-to-day variations.

An energy-based method for uniaxially compressed rocks and its implication

An energy-based method for uniaxially compressed rocks and its implication

EHIR: Energy-based Hierarchical Iterative Image Registration for Accurate PCB Defect Detection

Printed Circuit Board (PCB) Surface defect detection is crucial to ensure the quality of electronic products in manufacturing industry. Detection methods can be divided into non-referential and referential methods. Non-referential methods employ designed rules or learned data distribution without template images but are difficult to address the uncertainty and subjectivity issues of defects. In contrast, referential methods use templates to achieve better performance but rely on precise image registration. However, image registration is especially challenging in feature extracting and matching for PCB images with defective, reduplicated or less features. To address these issues, we propose a novel Energy-based Hierarchical Iterative Image Registration method (EHIR) to formulate image registration as an energy optimization problem based on the edge points rather than finite features. Our framework consists of three stages: Edge-guided Energy Transformation (EET), EHIR and Edge-guided Energy-based Defect Detection (EEDD). The novelty is that the consistency of contours contributes to aligning images and the difference is highlighted for defect location. Extensive experiments show that this method has high accuracy and strong robustness, especially in the presence of defect feature interference, where our method demonstrates an overwhelming advantage over other methods.

An adaptive energy-based sequential method for training PINNs to solve gradient flow equations

Recent developments in numerous scientific computing fields have been made possible by physics-informed neural networks (PINNs), a class of machine learning techniques that incorporate physical knowledge into neural networks. However, PINNs still encounter difficulties in simulating a kind of partial differential equations (PDEs), i.e. gradient flow equations of complex dynamic systems. Through analysis from the neural tangent kernel's (NTK) point of view, we find that directly training PINNs on the large temporal window may lead to failure. To tackle this problem, we utilize one of the gradient flow equations' significant physical properties, the energy dissipation law, to propose an adaptive energy-based sequential training method for PINNs. Specifically, the gradient of free energy is employed to determine the adaptive temporal steps of the training window, which also enhances the training efficiency. Numerical experiments are demonstrated among gradient flow equations including Cahn-Hilliard, heat conduction, and Allen-Cahn equations. Our method not only outperforms PINNs but also preserves the energy dissipation law, which shows the ability to solve complex long-time simulation problems of gradient flow systems.

Computational Modeling of Tool-Rock Frictional Contact With Anisotropic Damage

The mechanism of frictional contact is investigated between a blunt tool and quasi-brittle rocks with anisotropic damage. A recently developed anisotropic elasto-plastic-damage model is further validated using experimental results under monotonic and cyclic loadings for rocks and concrete. A finite element model of tool-rock frictional contact is validated by analytical results in the two asymptotic regimes of an elastoplastic rock. The mesh sensitivity is reduced using a fracture energy-based method for anisotropic damage. The tool-rock frictional contact is predominantly controlled by three dimensionless parameters: η\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\eta$$\\end{document}, ξ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\xi$$\\end{document}, and ζ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\zeta$$\\end{document}, which characterize elastoplasticity, brittleness, and anisotropic damage, respectively. The newly introduced damage coefficient ζ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\zeta$$\\end{document} controls the ratio of damage in different directions. As the elastoplastic parameter η\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\eta$$\\end{document} increases with more plastic deformation, the dimensionless average contact stress Π~\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\widetilde{\\Pi }}$$\\end{document} increases and then slightly varies before stabilizing. As the brittleness number ξ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\xi$$\\end{document} increases in a more brittle mode, the contact stress Π~\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\widetilde{\\Pi }}$$\\end{document} generally decreases. When the damage coefficient ζ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\zeta$$\\end{document} decreases from isotropic to anisotropic damage, the contact stress Π~\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\widetilde{\\Pi }}$$\\end{document} generally increases. The magnitudes of the average contact stress in numerical modeling are closer to experimental results when considering anisotropic damage.

Energy-based seismic design method for seismically isolated structures with friction pendulum systems

Friction pendulum systems (FPSs) serve as effective isolators that can help ensure adequate seismic performance of structures against significant aftershocks. A commonly used method for FPS design is the equivalent static analysis method outlined in ASCE 7-22. However, this method risks underestimating the seismic performance of FPSs due to its inability to account for energy dissipation from multiple hysteresis loops exceeding two cycles. Moreover, it is not well-suited for designing FPSs with conical friction surfaces, which exhibit nonlinear elastic behavior and hysteresis. A straightforward solution to address these issues is to apply an energy-based seismic design method. In this study, we propose and analytically validate one such method tailored for seismically isolated structures employing FPSs. Our analysis considers FPSs with both spherical (ISF) and conical (ICF) friction surfaces. For ISFs, the design follows the energy-based seismic design method proposed by Akiyama, whereas, for ICFs, it is based on the energy input to the device and its geometric approximation to an ISF. The conclusions of this study are summarized as follows: (1) the proposed energy-based design method effectively predicts the maximum FPS deformation, facilitating a rough estimation of FPS size; (2) because the design assumes symmetrical seismic response of the isolation layer in both the positive and negative directions and the friction surfaces of the ISF and ICF are geometrically similar, in the proposed method, the maximum deformation of an isolation layer with a natural period greater than 2 s depends solely on the friction coefficient, regardless of the friction surface type; (3) to satisfy the relevant conditions, the appropriate FPS design range is determined through nonlinear analysis. The results of this study demonstrate that the proposed energy-based seismic design method provides a reliable and effective means of designing FPSs for enhanced seismic performance.

Determination of dynamic collapse limit states using the energy-based method for multi-story RC frames subjected to column removal scenarios

The Alternative Load Path method is widely adopted to perform progressive collapse analyses for reinforced concrete (RC) frame structures subjected to column removal scenarios. Considering that the progressive collapses are usually dynamic phenomena, nonlinear dynamic analyses can yield the most accurate results. However, the associated computational cost is high. Alternatively, the energy-based method (EBM) can be employed to efficiently calculate the maximum dynamic responses. Unfortunately, collapse or dynamic limit states (DLS) cannot be directly determined when the EBM is adopted for the progressive collapse analyses, because it is essentially a quasi-static approach. To tackle this issue, in this paper an EBM-Intersection Point-based (EBM-IP-based) method is proposed to effectively determine the DLS. The method is considering that the dynamic instability will occur once the dynamic capacity curve exceeds the corresponding static capacity curve. Hence, it is possible to determine the DLS by making a correction on the intersection point of the aforementioned two curves without performing nonlinear dynamic time-history analyses. Compared with the other energy components, the kinetic energy at the moment of the peak displacement response is relatively small and the effectiveness of the EBM is verified. Through both static pushdown analyses and incremental dynamic analyses for 48 column removal scenarios of six different multi-story RC frames, the effectiveness of the proposed EBM-IP-based method is verified and the associated coefficients are calibrated. Statistical information and (joint) probabilistic density functions (PDF) of the coefficients are identified. Further, stochastic analyses are carried out in order to quantify the model uncertainties when adopting the EBM-IP-based method. On the basis of all the calculation results, a Gumbel distribution is adopted to capture the PDF of the model uncertainty in relation to displacements, while a lognormal distribution is adopted to fit the PDF of the model uncertainty in terms of resistances. Moreover, a multi-variate Gaussian distribution model with two components is employed to fit the joint PDF.

PiCO+: Contrastive Label Disambiguation for Robust Partial Label Learning.

Partial label learning (PLL) is an important problem that allows each training example to be labeled with a coarse candidate set with the ground-truth label included. However, in a more practical but challenging scenario, the annotator may miss the ground-truth and provide a wrong candidate set, which is known as the noisy PLL problem. To remedy this problem, we propose the PiCO+ framework that simultaneously disambiguates the candidate sets and mitigates label noise. Core to PiCO+, we develop a novel label disambiguation algorithm PiCO that consists of a contrastive learning module along with a novel class prototype-based disambiguation method. Theoretically, we show that these two components are mutually beneficial, and can be rigorously justified from an expectation-maximization (EM) algorithm perspective. To handle label noise, we extend PiCO to PiCO+, which further performs distance-based clean sample selection, and learns robust classifiers by a semi-supervised contrastive learning algorithm. Beyond this, we further investigate the robustness of PiCO+ in the context of out-of-distribution noise and incorporate a novel energy-based rejection method for improved robustness. Extensive experiments demonstrate that our proposed methods significantly outperform the current state-of-the-art approaches in standard and noisy PLL tasks and even achieve comparable results to fully supervised learning.

Multi-scale concurrent topology optimization of cellular structures with multiple microstructures for minimizing dynamic response in the time domain

This paper proposes a concurrent topology optimization method for optimizing the structures that are periodically filled with multiple microstructures excited by dynamic response in the time domain. At the macroscale, different microstructures are considered as different materials. To generate the various distribution of different microstructures, a multi-material interpolation approach based on the Solid Isotropic Material with Penalization (SIMP) is integrated. At the microscale, the energy-based homogenization method (EBHM) is employed to determine the macroscopic effective properties of the microstructures. All macroscale elements with identical material are represented by a distinct microstructure. For the proportional damping model, the HHT-α is implemented as a time integration technique to obtain the dynamic response of multi-scale and assumed multi-material structures. The objective function of the topology optimization problem is to minimize the mean dynamic compliance. Combined differentiate-then-discretize method with the adjoint variable method, the sensitivity analysis applies the gradient-based Zhang-Paulino-Ramos Jr. (ZPR) algorithm or Method of Moving Asymptotes (MMA) to update the design variables under multiple constraints in the time and space-discretized system. The proposed approach is numerically performed through 2D and 3D examples to demonstrate its effectiveness.

Inertia Energy-Based Required Capacity Calculation of BESS for Achieving Carbon Neutrality in Korean Power System

Frequency response performance in power systems is becoming vulnerable due to the transition toward the higher penetration of renewable energy such as achieving carbon neutrality. In particular, reducing power system inertia energy as the asynchronous generation increases could result in violating the frequency stability constraint when a disturbance occurs in the power systems. In order to control the rapidly fluctuating frequency of the power system with low inertia, it is necessary to introduce fast frequency response resources such as a Battery Energy Storage System (BESS). This paper proposes a method to calculate the required capacity of BESS for compensating the frequency control performance of the power system using inertia energy. For calculating the required capacity of BESS, the inertia energy in the critical power system, where frequency control performance marginally satisfies frequency stability constraint, should be calculated. Also, the inertia energy in the evaluated power system having deficit inertia energy should be calculated. By comparing power systems that respond with different dynamics when the same disturbance occurs, the proposed calculation corresponds to the ratio of inertia energy deficiency based on critical power system inertia energy within the power imbalance. Through various case studies employing Korean power systems, the effectiveness of the inertia energy-based calculation method for the required BESS is verified by the fact that the BESS integrated power system marginally satisfies the frequency stability constraint. In these study cases, it is found that the instant response of BESS is very effective for compensating the frequency control performance of the low inertia power system. By applying the proposed method, it is also found that about 840 MW of BESS can achieve carbon neutrality in the Korean power system.