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DOE OSTI2021

Model America - data and models of every U.S. building

The 5-year goal of the 'Model America' concept was to generate a model of every building in the United States. This data repository delivers on that goal. Oak Ridge National Laboratory (ORNL) has developed the Automatic Building Energy Modeling (AutoBEM) software suite to process multiple types of data, extract building-specific descriptors, generate building energy models, and simulate them on High Performance Computing (HPC) resources. For more information, see AutoBEM-related publications (bit.ly/AutoBEM). There were 125,714,640 buildings detected in the United States and this dataset contains 122,930,327 (97.8%) buildings which resulted in a successful simulation. Future, annual updates have been proposed that may include additional buildings, data improvements, or other algorithmic enhancements. This dataset of 122.9 million buildings includes: Models (state_county.zip) - OpenStudio (v3.1.0) and EnergyPlus (v9.4) building energy models. Please note that the download requires the free Globus Connect Personal (https://www.globus.org/globus-connect-personal); Each model has approximately 3,000 building input descriptors that can be extracted. Please see the EnergyPlus(v9.4) 2,784-page Input/Output Reference Guide (https://energyplus.net/sites/all/modules/custom/nrel_custom/pdfs/pdfs_v9.4.0/InputOutputReference.pdf) for everything that can be retrieved or simulated from these models. These models were derived from the following metadata, which is not included in this dataset: 1. ID - unique building ID 2. County - county name 3. State - state name 4. CZ - ASHRAE Climate Zone designation 5. Clim_Zone - text label of climate zone 6. est_year - estimated year of construction 7. est_commercial - estimated building type (0=residential, 1=commercial) 8. Centroid - building center location in latitude/longitude (from Footprint2D) 9. Footprint2D - building polygon of 2D footprint (lat1/lon1_lat2/lon2_...) 10. Height - building height (meters) 11. Area2D - footprint area (ft2) 12. BuildingType - DOE prototype building designation (IECC=residential) as implemented by OpenStudio-standards 13. WWR_surfaces - percent of each facade (pair of points from Footprint2D) covered by fenestration/windows (average 14.5% for residential, 40% for commercial buildings) 14. NumFloors - number of floors (above-grade) 15. Area - estimate of total conditioned floor area (ft2) 16. Standard - building vintage. These models are made free and openly available in hopes of stimulating any simulation-informed use case. Data is provided as-is with no warranties, express or implied, regarding fitness for a particular purpose. We wish to thank our sponsors which include Oak Ridge National Laboratory (ORNL) Laboratory Directed Research and Development (LDRD), U.S. Dept. of Energy's (DOE) Building Technologies Office (BTO), Office of Electricity (OE), Biological and Environmental Research (BER), and National Nuclear Security Administration (NNSA). This research used resources of the Argonne Leadership Computing Facility, which is a DOE Office of Science User Facility supported under Contract DE-AC02-06CH11357. Please cite as: New, Joshua R., Adams, Mark, Bass, Brett, Berres, Anne, and Clinton, Nicholas (2021). 'Model America - data and models of every U.S. building. [Data set].' Constellation, doi.ccs.ornl.gov/ui/doi/339, April 14, 2021

24 POWER TRANSMISSION AND DISTRIBUTION↗

DOE OSTI2023

Mathematical nuances of Gaussian process-driven autonomous experimentation

Abstract The fields of machine learning (ML) and artificial intelligence (AI) have transformed almost every aspect of science and engineering. The excitement for AI/ML methods is in large part due to their perceived novelty, as compared to traditional methods of statistics, computation, and applied mathematics. But clearly, all methods in ML have their foundations in mathematical theories, such as function approximation, uncertainty quantification, and function optimization. Autonomous experimentation is no exception; it is often formulated as a chain of off-the-shelf tools, organized in a closed loop, without emphasis on the intricacies of each algorithm involved. The uncomfortable truth is that the success of any ML endeavor, and this includes autonomous experimentation, strongly depends on the sophistication of the underlying mathematical methods and software that have to allow for enough flexibility to consider functions that are in agreement with particular physical theories. We have observed that standard off-the-shelf tools, used by many in the applied ML community, often hide the underlying complexities and therefore perform poorly. In this paper, we want to give a perspective on the intricate connections between mathematics and ML, with a focus on Gaussian process-driven autonomous experimentation. Although the Gaussian process is a powerful mathematical concept, it has to be implemented and customized correctly for optimal performance. We present several simple toy problems to explore these nuances and highlight the importance of mathematical and statistical rigor in autonomous experimentation and ML. One key takeaway is that ML is not, as many had hoped, a set of agnostic plug-and-play solvers for everyday scientific problems, but instead needs expertise and mastery to be applied successfully. Graphical abstract

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2025

Computational Algorithms for Unit Commitment with AC Power Flows (Final Report)

Security-constrained unit commitment (SCUC) is a key component in power system operations. When AC power flow constraints are considered in the SCUC model (AC-SCUC), the problem becomes extremely difficult due to its discrete and non-convex nature, as described in “Grid Optimization Competition Challenge 3 Problem Formulation (GOCC)”. There are four main challenges: (i) Discrete decisions regarding unit online/offline status and start-up/shut-down procedures for every single unit. The number of discrete decision variables increases considerably when a system integrates multiple generators; (ii) Configuration-based combined-cycle formulations, and multi-commodity models that include ramping products, spin/non-spin products, and regulation up/down products. The combined-cycle units introduce additional discrete decision variables and auxiliary service products further complicate the model by connecting multi-commodity products’ continuous and discrete variables; (iii) SCUC models with AC power flow constraints are far more complex due to massive bilinear terms in the large-scale nonlinear power balance equations. The nonlinear power balance equations are further complicated by the discrete step control variables of shunts; (iv) N − 1 contingency analysis. The size of the model increases linearly with the number of contingencies considered, greatly increasing the size of the optimization model. Accordingly, there is an emergent need to develop a robust algorithm capable of deriving a high-quality solution in a short time and passing through contingency tests simultaneously. In this project, we explore innovative techniques to address this challenging problem by integrating advanced polyhedral theory, approximation methods, relaxation strategies, decomposition techniques, and parallel computing. Each technique approaches the problem from a different perspective, leveraging its specific strengths to tackle distinct challenges. Each individual method has demonstrated its effectiveness in the PI’s previous research. Their integration is expected to significantly reduce the computational time required to solve the proposed complex problem. Successful completion of this project has the potential to transform the industry by enhancing optimization solvers capable of handling large-scale day-ahead energy market clearing models within strict time constraints, while incorporating AC power flow constraints. This advancement will lead to reduced overall generation costs and, consequently, increased social welfare.

29 ENERGY PLANNING, POLICY, AND ECONOMY↗

DOE OSTI2023

Domain Adaptive Graph Neural Networks for Constraining Cosmological Parameters Across Multiple Data Sets

State of the art astronomical simulations have provided datasets which enabled the training of novel deep learning techniques for constraining cosmological parameters. However, differences in subgrid physics implementation and numerical approximations among simulation suites lead to differences in simulated datasets, which pose a hard challenge when trying to generalize across diverse data domains and ultimately when applying models to observational data. Recent work reveals deep learning algorithms are able to extract more information from complex cosmological simulations than summary statistics like power spectra. We introduce Domain Adaptive Graph Neural Networks (DA-GNNs), trained on CAMELS data, inspired by CosmoGraphNet (Villanueva-Domingo et al 2023). By utilizing GNNs, we can capitalize on their capacity to capture both astrophysical and topological features of galaxy distributions. Mixing these capabilities with domain adaptation techniques such as Maximum Mean Discrepancy (MMD), which enable extraction of domain-invariant features, our framework demonstrates enhanced accuracy and robustness. We present experimental results, including the alignment of distributions across domains through data visualization. These findings suggest that DA-GNNs are an efficient way of extracting domain independent cosmological information, a vital step toward robust deep learning for real cosmic survey data.

79 ASTRONOMY AND ASTROPHYSICS↗

DOE OSTI2021

An Improved Algorithm for Estimating Surface Shortwave Radiation: Preliminary Evaluation With MODIS Products

Cloud parameters, as key inputs in radiative transfer algorithms, have a critical impact on surface shortwave radiation (SSR) computation. By introducing a parameterization of cloud transmittance and reflectance, based on radiative transfer simulations, this study improves the accuracy of an existing physically based model which severely underestimates SSR under thick cloud conditions. The cloud parameterization adopts the single-layer cloud model and simulates cloud transmittances and reflectances by varying cloud optical thickness, cloud particle size, and solar zenith angle. The revised model is applied to estimate instantaneous SSR using Moderate-resolution Imaging Spectroradiometer (MODIS) atmospheric and land products. The retrieved SSR is evaluated against observation data from 41 Baseline Surface Radiation Network (BSRN) stations and is also compared with the MODIS official SSR product. The root mean square error (RMSE) of the estimated instantaneous radiation is approximately 52 and 98 W m -2 under clear-sky and all-sky conditions, respectively. The accuracy of the improved parameterization is higher than that of the original model, and there is no obvious underestimation of SSR in the case of high cloud optical thickness. Therefore, the new algorithm improves the accuracy of SSR estimates in the presence of thick clouds. Retrievals with the improved model also achieve higher accuracy than the MODIS official SSR product (MCD18A1). To conclude, the reliable performance of the scheme at most BSRN stations illustrates that the improved model can be used to map SSR on a global scale.

42 ENGINEERING↗

DOE OSTI2020

Design, Development and Testing Plan for Energy Efficiency Algorithms Related to Building-Integrated Cooling, Heating, and Power Systems

Building-integrated cooling, heating and power (CHP) systems are more efficient than conventional systems at providing local power and thermal energy, and favorable fuel prices are bound to spur their increased adoption. It is estimated that the total CHP technical potential in the commercial building sector is approximately 76 GW, but only a small fraction of it is actually deployed. Advanced control and monitoring systems that realize the full potential of these CHP systems can transform this capacity into an asset for both the owner and the grid and also accelerate the adoption of these systems in the building sector. However, to realize the full benefit of the CHP systems, we must ensure persistence of energy efficient operations. It has been demonstrated by several studies that as much as 30% of building energy use is wasted. Much of the waste results from the inability to anticipate load variations (design day vs. non-design day operations; diurnal variation of loads), identify improper operation, and efficiently operate, control and maintain building systems. Moreover, many buildings, even those with adequate control infrastructure, lack knowledgeable staff to operate and maintain energy systems. Operators of buildings with CHP systems will constantly need to make tradeoffs (economic dispatch) between onsite power generation and purchasing power, between onsite heat production from distributed generation versus heating/cooling from conventional systems, and different passive and active load management measures. Much of the inefficiency in the current building operations can be eliminated by use of automated performance monitoring (PM), real-time commissioning verification (CxV) and automated fault detection and diagnostic (AFDD) tools. Automation can help system operators make intelligent decisions. Remote and continuous monitoring of system conditions and performance will enable better management and integration of CHP with existing building systems. Continuous PM, real-time CxV and AFDD could alleviate burdens for operations staff, enhance operations and maintenance (O&M), and improve reliability of building and CHP systems. To address the O&M challenges and to provide a means to maximize the rate-of-return of building-integrated CHP systems, the Building Technologies Office (BTO) within the U.S. Department of Energy’s (DOE’s) Office of Energy Efficiency and Renewable Energy (EERE) initiated a project to design, develop, and field test a VOLTTRON™-based supervisory controller and associated open-source algorithms. These algorithms will ensure real-time optimal operation of a building-integrated CHP system, support electric grid reliability, and lead to achieving the goal of a clean, efficient, reliable and affordable next-generation integrated energy systems. This report lists the components for which PM, real-time CxV and AFDD algorithms will be developed, how the algorithms will be tested, and metrics that will be used to validate the algorithms and easy of deployment. Deployment of these algorithms in the field will result in savings reduction in energy consumption of between 10% and 20%.

32 ENERGY CONSERVATION, CONSUMPTION, AND UTILIZATI↗

DOE OSTI2021

Hydro-chemo-mechanical phase field formulation for corrosion induced cracking in reinforced concrete

Steel corrosion induced cracking in reinforced concrete structures is often caused by volumetric straining at the steel-concrete interface. This volumetric strain is mainly controlled by the rate of generation of corrosion product, which depends on the diffusion of Cl{sup −} and dissolved O{sub 2} as well as the chemical reaction at the steel-concrete interface. While many numerical modeling techniques have been proposed to tackle this complex phenomenon, most have been limited to using staggered solution schemes, where the diffusion, chemical reaction, mechanical strain, and crack fields are solved for separately. Unfortunately, such approaches introduce numerical errors that can lead to unrealistic predictions. Therefore, a coupled solution, is needed to remove the formulation induced errors and understand this complex phenomenon. To provide a unifying solution framework for corrosion induced cracking in reinforced concrete, a phase field formulation is presented and implemented with FEM algorithms. The governing equations are derived from the system's free energy and Fick's law of diffusion. The proposed formulation was first validated in 1D by comparing the modeling results with the semi-analytical solution approximated via Laurent series to avoid singularities in the complex domain. 2D plane strain solutions to the embedded rebar undergoing corrosion induced cracking with pre-cracks are also presented and compared with the approximated solutions from linear elastic fracture mechanics theory (LEFM). The modeling of crack propagation enabled by the proposed formulation is also presented along with parametric studies to reveal the roles of the fracture toughness and permeability of the concrete on the corrosion induced cracking. A case study was also conducted on H-pile steel with concrete jackets to demonstrate the feasibility of modeling corrosion induced fracture in reinforced concrete structures with complex steel-concrete interfaces.

36 MATERIALS SCIENCE↗

DOE OSTI2022

Generalized quasiharmonic approximation via space group irreducible derivatives

The quasiharmonic approximation (QHA) is the simplest nontrivial approximation for interacting phonons under constant pressure, bringing the effects of anharmonicity into temperature-dependent observables. Nonetheless, the QHA is often implemented with additional approximations due to the complexity of computing phonons under arbitrary strains, and the generalized QHA, which employs constant stress boundary conditions, has not been completely developed. In this work we formulate the generalized QHA, providing a practical algorithm for computing the strain state and other observables as a function of temperature and true stress. We circumvent the complexity of computing phonons under arbitrary strains by employing irreducible second-order displacement derivatives of the Born-Oppenheimer potential and their strain dependence, which are efficiently and precisely computed using the lone irreducible derivative approach. We formulate two complementary strain parametrizations: a discretized strain grid interpolation and a Taylor series expansion in symmetrized strain. We illustrate our approach by evaluating the temperature and pressure dependence of select elastic constants and the thermal expansion in thoria (ThO 2 ) using density functional theory with three exchange-correlation functionals. The QHA results are compared to our measurements of the elastic constant tensor using time-domain Brillouin scattering and inelastic neutron scattering. Our irreducible derivative approach simplifies the implementation of the generalized QHA, which will facilitate reproducible, data-driven applications.

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗

DOE OSTI2020

Low-Memory, Discrete Ordinates, Discontinuous Galerkin Methods for Radiative Transport

The discrete ordinates discontinuous Galerkin (SN-DG) method is a well-established and practical approach for solving the radiative transport equation. In this paper, we study a low-memory variation of the upwind SN-DG method. The proposed method uses a smaller finite element space that is constructed by coupling spatial unknowns across collocation angles, thereby yielding an approximation with fewer degrees of freedom than the standard method. Like the original SN-DG method, the low-memory variation still preserves the asymptotic diffusion limit and maintains the characteristic structure needed for mesh sweeping algorithms. While we observe second-order convergence in the scattering dominated, diffusive regime, the low-memory method is in general only first-order accurate. To address this issue, we use upwind reconstruction to recover second-order accuracy. Finally, for both methods, numerical procedures based on upwind sweeps are proposed to reduce the system dimension in the underlying Krylov solver strategy.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2024

Analysis of a Computational Framework for Bayesian Inverse Problems: Ensemble Kalman Updates and MAP Estimators under Mesh Refinement

This paper analyzes a popular computational framework to solve infinite-dimensional Bayesian inverse problems, discretizing the prior and the forward model in a finite-dimensional weighted inner product space. We demonstrate the benefit of working on a weighted space by establishing operator-norm bounds for finite element and graph-based discretizations of Matérn-type priors and deconvolution forward models. For linear-Gaussian inverse problems, we develop a general theory to characterize the error in the approximation to the posterior. We also embed the computational framework into ensemble Kalman methods and MAP estimators for nonlinear inverse problems. Furthermore, our operator-norm bounds for prior discretizations guarantee the scalability and accuracy of these algorithms under mesh refinement.

Bayesian inverse problem↗

DOE OSTI2024

Recent Advances of PyROS: A Pyomo Solver for Nonconvex Two-Stage Robust Optimization in Process Systems Engineering

The document presents recent algorithmic and implementation advances of the two-stage robust optimization (RO) solver PyROS, and a benchmarking study which demonstrates the utility of PyROS for two-stage RO problems. The advances include extensions of the scope of PyROS to models with uncertain variable bounds, improvements to the initializations of the subproblems used by the underlying cutting set algorithm, and extensions of the uncertainty set interfaces. The benchmarking study is performed on a library of over 8,500 instances, with variations in the nonlinearities, degree-of-freedom partitioning, uncertainty sets, and polynomial decision rule approximations. An amine-based CO2 capture case study is presented to demonstrate the utility of PyROS for large-scale process models. Overall, the results highlight the effectiveness of PyROS for obtaining robust solutions to optimization problems with uncertain equality constraints.

Sherman, Jason↗

DOE OSTI2023

DeepSurveySim: Simulation Software and Benchmark Challenges for Astronomical Observation Scheduling

Modern astronomical surveys have multiple competing scientific goals. Optimizing the observation schedule for these goals presents significant computational and theoretical challenges, and state-of-the-art methods rely on expensive human inspection of simulated telescope schedules. Automated methods, such as reinforcement learning, have recently been explored to accelerate scheduling. However, there do not yet exist benchmark data sets or user-friendly software frameworks for testing and comparing these methods. We present DeepSurveySim -- a high-fidelity and flexible simulation tool for use in telescope scheduling. DeepSurveySim provides methods for tracking and approximating sky conditions for a set of observations from a user-supplied telescope configuration. We envision this tool being used to produce benchmark data sets and for evaluating the efficacy of ground-based telescope scheduling algorithms, particularly for machine learning algorithms that would suffer in efficacy if limited to real data for training.We introduce three example survey configurations and related code implementations as benchmark problems that can be simulated with DeepSurveySim.

46 INSTRUMENTATION RELATED TO NUCLEAR SCIENCE AND ↗

DOE OSTI2023

Recent Advances in PyROS: The Pyomo Solver for Two-Stage Nonconvex Robust Optimization

The slides present recent algorithmic and implementation advances of the two-stage robust optimization (RO) solver PyROS, and a benchmarking study which demonstrates the utility of PyROS for two-stage RO problems. The advances include extensions of the scope of PyROS to models with uncertain variable bounds, improvements to the initializations of the subproblems used by the underlying cutting set algorithm, and extensions of the uncertainty set interfaces. The benchmarking study is performed on a library of over 8,500 instances, with variations in the nonlinearities, degree-of-freedom partitioning, uncertainty sets, and polynomial decision rule approximations. Overall, the results highlight the effectiveness of PyROS for obtaining robust solutions to optimization problems with uncertain equality constraints.

Sherman, Jason↗

DOE OSTI2023

Recent Advances in PyROS: The Pyomo Solver for Two-Stage Nonconvex Robust Optimization

Sherman, Jason↗

DOE OSTI2021

HAIMOS Ensemble Forecasts for Intra-day and Day- Ahead GHI, DNI and Ramps

The objective of this research is to develop a hybrid physics-based/data-driven forecast model to improve direct normal and global horizontal irradiance (DNI and GHI) prediction for horizons ranging from 1 to 72 hours. Project objectives also address key gaps in state-of-the-art solar forecasting: accurate probabilistic solar forecasts and the forecasting of large irradiance ramps (ramp onset and magnitude). The proposed model ensembles Numerical Weather Prediction (NWP) forecasts, determinist physics-based algorithms, and new-generation cloud cover products (high-resolution rapid refresh satellite images and Large Eddy Simulations). The result is the Hybrid Adaptive Input Model Objective Selection (HAIMOS) ensemble model. HAIMOS blends state of the art machine learning methodologies with physics-based models for cloud cover and cloud optical depth forecasts. The technical activities followed a two-pronged strategy. First, the preprocessing of data, the selection of inputs to the nonlinear approximators, the type of approximator and objective functions, and post-processing ensembling techniques included in HAIMOS were all optimized adaptively to find the best model for a specific goal (reduce DNI/GHI forecast error, improve the prediction of ramp onset, etc.). Second, a large effort was put in improving cloud identification and the forecast of cloud cover and cloud optical depth. To this end, new-generation cloud parametrization products were developed in this work. These include improved algorithms to assist in cloud identification, cloud classification and cloud parametrization from satellite images – three key factors in the accuracy of 1 to 6-hours irradiance forecasts and prediction of ramp onset. Furthermore, we also included cloud information extracted from high resolution rapid refresh satellite images (GOES-16) and Large Eddy Simulations (LES). LES was used to model the atmosphere in detail over locations of interest and produce cloud optical depth forecasts. Once these data streams were validated, they were used as input data to the HAIMOS forecast. The model was developed using data from several climatologically distinct locations with potential for high solar penetration. In the last year of the project, we conducted a validation campaign according to the guidelines stipulated by the Topic Area 1 project as described in the FOA. This effort brings, for the first time, proven machine-learning methodologies for generating state-of-the-art solar forecasts interweaved with detailed physics-based models for cloud detection, and cloud optical depth forecasts. HAIMOS will generate accurate irradiance probabilistic forecast to assist in reducing solar generation prediction error. Globally optimized solar forecast models are more likely to impact solar energy stakeholders. The goal of this project was to increase the state-of-the-art forecast skill from their present values of 10 to 35%. At the end of the project, we achieved between 30% and 50% forecast skill across a wide range of horizons for both GHI and DNI.

14 SOLAR ENERGY↗

DOE OSTI2023

Direct stellarator coil optimization for nested magnetic surfaces with precise quasi-symmetry

We present a robust optimization algorithm for the design of electromagnetic coils that generate vacuum magnetic fields with nested flux surfaces and precise quasi-symmetry. The method is based on a bilevel optimization problem, where the outer coil optimization is constrained by a set of inner least squares optimization problems whose solutions describe magnetic surfaces. The outer optimization objective targets coils that generate a field with nested magnetic surfaces and good quasi-symmetry. The inner optimization problems identify magnetic surfaces when they exist, and approximate surfaces in the presence of magnetic islands or chaos. We show that this formulation can be used to heal islands and chaos, thus producing coils that result in magnetic fields with precise quasi-symmetry. We show that the method can be initialized with coils from the traditional two-stage coil design process, as well as coils from a near-axis expansion optimization. We present a numerical example where island chains are healed and quasi-symmetry is optimized up to surfaces with aspect ratio 6. Another numerical example illustrates that the aspect ratio of nested flux surfaces with optimized quasi-symmetry can be decreased from 6 to approximately 4. In conclusion, the last example shows that our approach is robust and a cold-start using coils from a near-axis expansion optimization.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗

DOE OSTI2023

Bayesian projection pursuit regression

In projection pursuit regression (PPR), a univariate response variable is approximated by the sum of $M$ “ridge functions,” which are flexible functions of one-dimensional projections of a multivariate input variable. Traditionally, optimization routines are used to choose the projection directions and ridge functions via a sequential algorithm, and $M$ is typically chosen via cross-validation. Here, we introduce a novel Bayesian version of PPR, which has the benefit of accurate uncertainty quantification. To infer appropriate projection directions and ridge functions, we apply novel adaptations of methods used for the single ridge function case ($M$=1), called the Bayesian Single Index Model; and use a Reversible Jump Markov chain Monte Carlo algorithm to infer the number of ridge functions $M$. We evaluate the predictive ability of our model in 20 simulated scenarios and for 23 real datasets, in a bake-off against an array of state-of-the-art regression methods. Finally, we generalize this methodology and demonstrate the ability to accurately model multivariate response variables. Its effective performance indicates that Bayesian Projection Pursuit Regression is a valuable addition to the existing regression toolbox.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2024

Crack opening calculation in phase-field modeling of fluid-filled fracture: A robust and efficient strain-based method

The phase-field method has become popular for the numerical modeling of fluid-filled fractures, thanks to its ability to represent complex fracture geometry without algorithms. However, the algorithm-free representation of fracture geometry poses a significant challenge in calculating the crack opening (aperture) of phase-field fracture, which governs the fracture permeability and hence the overall hydromechanical behavior. Although several approaches have been devised to compute the crack opening of phase-field fracture, they require a sophisticated algorithm for post-processing the phase-field values or an additional parameter sensitive to the element size and alignment. Here, we develop a novel method for calculating the crack opening of fluid-filled phase-field fracture, which enables one to obtain the crack opening without additional algorithms or parameters. Here we transform the displacement-jump-based kinematics of a fracture into a continuous strain-based version, insert it into a force balance equation on the fracture, and apply the phase-field approximation. Through this procedure, we obtain a simple equation for the crack opening which can be calculated with quantities at individual material points. We verify the proposed method with analytical and numerical solutions obtained based on discrete representations of fractures, demonstrating its capability to calculate the crack opening regardless of the element size or alignment.

58 GEOSCIENCES↗