Data-driven multiscale finite-element method using deep neural network combined with proper orthogonal decomposition

Application of the multiscale finite element method to groundwater flow in heterogeneous porous media

We present an enhancement of the multiscale finite element method (MsFEM) by combining it with the hp-adaptive FEM. Such a discretization-based homogenization technique is a versatile tool for modeling heterogeneous materials with fast oscillating elasticity coefficients. No assumption on periodicity of the domain is required. In order to avoid direct, so-called overkill mesh computations, a coarse mesh with effective stiffness matrices is used and special shape functions are constructed to account for the local heterogeneities at the micro resolution. The automatic adaptivity (hp-type at the macro resolution and h-type at the micro resolution) increases efficiency of computation. In this paper details of the modified MsFEM are presented and a numerical test performed on a Fichera corner domain is presented in order to validate the proposed approach.

In many applications the resolution of small-scale heterogeneities remains a significant hurdle to robust and reliable predictive simulations. In particular, while material variability at the mesoscale plays a fundamental role in processes such as material failure, the resolution required to capture mechanisms at this scale is often computationally intractable. Multiscale methods aim to overcome this difficulty through judicious choice of a subscale problem and a robust manner of passing information between scales. One promising approach is the multiscale finite element method, which increases the fidelity of macroscale simulations by solving lower-scale problems that produce enriched multiscale basis functions. In this study, we present the first work toward application of the multiscale finite element method to the nonlocal peridynamic theory of solid mechanics. This is achieved within the context of a discontinuous Galerkin framework that facilitates the description of material discontinuities and does not assume the existence of spatial derivatives. Analysis of the resulting nonlocal multiscale finite element method is achieved using the ambulant Galerkin method, developed here with sufficient generality to allow for application to multiscale finite element methods for both local and nonlocal models that satisfy minimal assumptions. We conclude with preliminary results on a mixed-locality multiscale finite element method in which a nonlocal model is applied at the fine scale and a local model at the coarse scale.

Accurate prediction of endpoint carbon at the dynamic control stage in the converter is crucial for achieving smelting targets. Currently, there are two main methods for converter endpoint prediction: the data-driven method and the mechanism-based method. Data-driven methods exhibit high accuracy but are vulnerable to data quality variations and lack interpretability. Mechanism-based methods provide great interpretability but face challenges in precisely identifying key parameters in the mechanism formula. Inspired by the design concept of physics-informed neural networks (PINNs), an integrated data-driven and mechanism-based method for endpoint carbon prediction in BOF (dmPINNs, data-driven and mechanism-based physics-informed neural networks) is proposed, which has four parts: feature extraction, mechanism-based calculation, data-driven prediction, and integrated prediction. We identify key parameters of the mechanism formula through the neural network to obtain the specified formula for each heat and supervise the training process of the neural network through the mechanism formula to ensure interpretability. Experimental results show that, within the ±0.012% error range, the hit rate of endpoint carbon content using dmPINNs improved by 5.23% compared with the traditional data-driven method and has greater robustness with the supervision of the mechanism formula.

Data-driven multiscale finite element method: From concurrence to separation

For a regional land subsidence problem, there are difficulties for the traditional finite element method to solve the regional groundwater flow equation involved. A new finite element method - multiscale finite element method (MsFEM) applied to resolve three dimensional groundwater flow problem- is introduced. Three advantages make the MsFEM very useful for groundwater flow simulation in a large region with highly heterogeneous porous media. One advantage is that this method can significantly save computing efforts by coarse grid mesh. The variation of parameters in an element is brought to the multiscale base functions. Second one is that this method can to a large extent avoid large aspect ratios of elements by combining aquifers and aquitards in one element. The third advantage is that the hydraulic heads in different depths of aquitards or aquifers could be rather accurately interpolated using the multiscale base functions and the nodal heads of the element by MsFEM. The third advantage is useful for the subsidence modeling. A numerical example is done to verify the advantages of MsFEM. Finally, the MsFEM is applied to solve the regional land subsidence model of Shanghai, which includes a three-dimensional groundwater flow mathematical model and a one-dimensional subsidence model. The traditional two-step approach is followed in this study, with the hydrodynamics of the pumped multiaquifer system first simulated by a 3-D groundwater flow model and the subsidence then computed with a 1-D subsidence model with the pore pressure field specified as an external distributed source of strength within the porous medium. The simulation period is from March, 1986 to December, 1998. Using MsFEM, the aquifer system in Shanghai is divided into six layers, which means one aquifer and one aquitard are in one layer with an average thickness of about 50 meters. Every element in MsFEM is subdivided into 7 sub-layers in the vertical direction and every horizontal layer in an element is divided into 4 hexahedral elements. So each element is subdivided into 28 hexahedral sub-cells, which allows the heterogeneity of aquifers and aquitards to be considered by multiscale base functions. The aquifers system in total is discretized into 19,996 elements with a total of 25,000 nodes. The hydraulic heads in any depth of aquitards or aquifers could be interpolated using the multiscale base functions and the nodal heads of the element. It is very useful for the settlement calculation for the aquitards or aquifers after hydraulic heads at nodes of elements are obtained by MsFEM. The application of MsFEM significantly saves computing efforts, effectively decreases the aspect ratios of elements and obtains good results, which proves the MsFEM can be applied to solve the real regional 3D groundwater flow problems, and it is an effective numerical method for regional land subsidence modeling.

This study analyzed the pressure patterns and local pressure of tall buildings with corner modifications (recessed and chamfered corner) using wind tunnel tests and proper orthogonal decomposition (POD). POD can distinguish pressure patterns by POD mode and more dominant pressure patterns can be found according to the order of POD modes. Results show that both recessed and chamfered corners effectively reduced wind-induced responses. Additionally, unique effects were observed depending on the ratio of corner modification. Tall building models with recessed corners showed fluctuations in the approaching wind flow in the first POD mode and vortex shedding effects in the second POD mode. With large corner modification, energy distribution became small in the first POD mode, which shows that the effect of the first POD mode reduced. Among building models with chamfered corners, vortex shedding effects appeared in the first POD mode, except for the model with the highest ratio of corner modifications. The POD confirmed that both recessed and chamfered corners play a role in reducing vortex shedding effects, and the normalized power spectral density peak value of modes showing vortex shedding was smaller than that of the building model with a square section. Vortex shedding effects were observed on the front corner surfaces resulting from corner modification, as with the side surface. For buildings with recessed corners, the local pressure on corner surfaces was larger than that of side surfaces. Moreover, the average wind pressure was effectively reduced to 88.42% and 92.40% in RE1 on the windward surface and CH1 on the side surface, respectively.

The multiscale finite element method (MsFEM) combined with conventional finite element method (FEM) is proposed to solve static magnetic field in the ribbon magnetic core. Firstly, a simple 2-dimensional electrostatic problem is used to introduce the MsFEM part. The results are compared to analytical method, as well as conventional FEM. Then, an example of magneto static problem is considered for a ribbon magnetic core built sheet by sheet. Conventional FEM and MsFEM are used to compute the magneto-static field by adopting scalar magnetic potential. Both magnetic potential and magnetic flux density on a certain path are compared. It is shown that the results obtained by MsFEM agrees well with the one from conventional FEM. Moreover, MsFEM combined with FEM is potentially a general strategy for multiscale modeling of ribbon magnetic cores, which can effectively reduce the computational cost.

The cycle-to-cycle variation of engine in-cylinder flow is critical for the improvement of performance for spark-ignition internal combustion engines. Proper orthogonal decomposition (POD), with its ability to extract the most energetic fluctuation structure, is widely used to analyze the in-cylinder flow and understand the variation of its evolution in different cycles. However, both of the two existing approaches to use POD for engine flow analysis encounter difficulties when applied for this purpose. Phase-dependent POD decomposes a data set in which all samples are taken at a certain engine phase (crank angle) from different cycles, but the POD results at neighboring engine phases do not necessarily evolve coherently. Phase-invariant POD, when applied to analyze tumble flow, stretches/compresses and interpolates the flow fields obtained at different engine phases onto the same grid, and this deformation means that phase-invariant POD results are no longer significant in energy sense. To overcome these difficulties, we propose an adaptation of conditional space-time POD to work with engine flow, with which the flow within a range of engine phases in each cycle is considered as one sample. It is shown that the low-order modes obtained with conditional space-time POD capture fluctuation structures that evolve coherently, and these results are compared and contrasted with those of the two existing POD approaches. A reduced-order model of the engine in-cylinder flow is constructed based on the partial sum of the modes and coefficients obtained from the conditional space-time POD, and it is shown that this new reduced-order model identifies structure that is both coherent spatially and temporally.

Conventional finite-element method for solving Helmholtz equation on highly heterogeneous media usually require very finely discretized mesh to accurately represent the medium property variations, and therefore is computationally expensive. We present a continuous Galerkin generalized multiscale finite element method for solving Helmholtz equation in media with variable velocity and mass density on coarse mesh. Instead of using polynomial basis functions, our multiscale method use multiscale functions obtained by multiplying the eigenfunctions from local spectral problems with an appropriately selected multiscale partition of unity to form the approximation space on the coarse mesh. These multiscale basis can efficiently capture the information of fine scale features of the medium without solving the problem on the fine mesh. We show by a numerical example that our multiscale method can greatly reduce the dimension of the coefficient matrix of Helmholtz equation system, as well as the computational time, with very small error.

When solving the 3D direct current resistivity logging problem with complex heterogeneous structures, the finite-element method requires the construction of a fine discretized mesh to accurately reflect the significant heterogeneity of the steel casing and the complexity of the underground structure. This fine mesh division will require a large amount of computing resources, making it difficult to solve in practice. The multiscale finite-element method can significantly reduce the size of the coefficient matrix, which greatly decreases the computational cost. However, the low-order multiscale finite-element method cannot accurately transfer the heterogeneous information of the fine mesh to the coarse mesh, which leads to numerical errors. In addition, imposing linear boundary conditions to construct multiscale basis functions may introduce resonance error. Consequently, we implement a 3D direct current resistivity logging forward algorithm based on high-order multiscale finite elements. On the one hand, we construct boundary and internal high-order multiscale basis functions by using arbitrary-order orthogonal polynomials within local cells and then linearly combine them to form complete high-order multiscale basis functions, ensuring the numerical solution’s approximation ability and convergence accuracy. On the other hand, we construct temporary high-order multiscale basis functions on extended local cells, combined with an oversampling strategy to mitigate the error problem caused by boundary resonance effects. We simulate three different electrode configurations: pole-pole, pole-dipole, and dipole-dipole. The results indicate that our new method can approximate the fine-scale reference solution on the coarse mesh with high accuracy and significantly reduced computational time when solving the linear system.

Extended multiscale finite element method based on polyhedral coarse grid elements for heterogeneous materials and structures

The jet in crossflow describes a unique flow characteristic, which involves the injection of a fluid jet into a turbulent boundary layer. This flow structure is significant for applications in vibration reduction, noise suppression, and cooling and heat transfer. In this study, a numerical model for jet-in-crossflow was established using Large Eddy Simulation, and the accuracy of simulation results was validated by comparison with experimental data. On the basis of maintaining the same jet exit flow rate, the study investigated the differences in fundamental flow characteristics, wall pressure fluctuations, and flow noise among jet-in-crossflow cases with circular, square, and elliptical orifices. The distribution of sound sources for the three cases was identified using vortex sound theory. The study shows that the elliptical orifice case did not develop a jet blockage effect similar to the other cases. All three orifice cases generated counter-rotating vortex pairs, although the vortex core positions varied. For the circular and square orifices, the root mean square pressure along the lower edge of the orifice exhibited a symmetric distribution, while the elliptical orifice case showed no clear symmetry. In the downstream region, spanning 2–10 orifice diameters, a stable dominant frequency was observed for all three cases, which is attributed to vortex transport. The elliptical orifice demonstrated significant noise reduction performance, with a noise reduction of 3–8 dB, mainly concentrated in the downstream region up to 10 orifice diameters. Using Proper Orthogonal Decomposition and Dynamic Mode Decomposition, the relationship between vortex transport and wall pressure fluctuations for the three cases was explained. The analysis revealed the fundamental reason behind the noise reduction capabilities of the elliptical orifice and the vortex structures responsible for generating the dominant frequency.

Shear-layer driven open cavity flows are known to exhibit strong self-sustained oscillations of the shear-layer. Over some range of the control parameters, a competition between two modes of oscillations of the shear layer can occur. We apply both Proper Orthogonal Decomposition and Dynamic Mode Decomposition to experimental two-dimensional two-components time and spaced velocity fields of an incompressible open cavity flow, in a regime of mode competition. We show that, although proper orthogonal decomposition successes in identifying salient features of the flow, it fails at identifying the spatial coherent structures associated with dominant frequencies of the shear-layer oscillations. On the contrary, we show that, as dynamic mode decomposition is devoted to identify spatial coherent structures associated with clearly defined frequency channels, it is well suited for investigating coherent structures in intermittent regimes. We consider the velocity divergence field, in order to identify spanwise coherent features of the flow. Finally, we show that both coherent structures in the inner-flow and in the shear-layer exhibit strong spanwise velocity gradients, and are therefore three-dimensional.

In this paper we propose local approximation spaces for localized model order reduction procedures such as domain decomposition and multiscale methods. Those spaces are constructed from local solutions of the partial differential equation (PDE) with random boundary conditions, yield an approximation that converges provably at a nearly optimal rate, and can be generated at close to optimal computational complexity. In many localized model order reduction approaches like the generalized finite element method, static condensation procedures, and the multiscale finite element method local approximation spaces can be constructed by approximating the range of a suitably defined transfer operator that acts on the space of local solutions of the PDE. Optimal local approximation spaces that yield in general an exponentially convergent approximation are given by the left singular vectors of this transfer operator [I. Babu\v{s}ka and R. Lipton 2011, K. Smetana and A. T. Patera 2016]. However, the direct calculation of these singular vectors is computationally very expensive. In this paper, we propose an adaptive randomized algorithm based on methods from randomized linear algebra [N. Halko et al. 2011], which constructs a local reduced space approximating the range of the transfer operator and thus the optimal local approximation spaces. The adaptive algorithm relies on a probabilistic a posteriori error estimator for which we prove that it is both efficient and reliable with high probability. Several numerical experiments confirm the theoretical findings.