Nonconforming Meshes Research Articles

Application of deep learning reduced-order modeling for single-phase flow in faulted porous media

AbstractOur research is positioned within the framework of subsurface resource utilization for sustainable economies. We concentrate on modeling the underground single-phase fluid flow affected by geological faults using numerical simulations. The study of such flows is characterized by strong uncertainites in the data defing the problem due to the difficulty of taking precise measurements in the subsoil. We aim to demonstrate the feasibility of a reduced order model that is both reliable and computationally efficient, thereby facilitating the incorporation of uncertainties. We account for the uncertainities of the properties of the rock and the geometry of the fault. The latter is achieved by using a radial basis function mesh deformation method. This approach benefits from a mixed-dimensional framework to model the rock matrix and faults as n and $${n-1}$$ n - 1 dimensional domains, allowing for non-conforming meshes. Our primary focus is on a reduced-order model capable of reproducing flow variables across the entire domain. We utilize the Deep Learning Reduced Order Model (DL-ROM), a nonintrusive neural network-based technique, and we compare it against the traditional Proper Orthogonal Decomposition (POD) method across various scenarios. The most relevant contributions of this work are: the proof of concept of the use of neural network for reduced order models for subsoil flow, dealing with non-affine problems and mixed dimensional domain. Additionally, we generalize an existing mesh deformation method for discontinuous deformation maps. Our analysis highlights the capability of reduced order model, highlighting DL-ROM’s capacity to expedite complex analyses with promising accuracy and efficiency, making multi-query analyses with various quantities of interest affordable.

Conforming mesh modeling of multi-physics effect on residual stress in multi-layer powder bed fusion process

The current research aims to predict the residual stress accumulation and evolution in the powder bed fusion processed multi-layer thin wall structures through a conforming mesh modeling approach. It involves the discrete element method (DEM) interfaced with the volume of fluid (VOF) method using computational fluid dynamics (CFD) coupled with the finite element method (FEM). The conforming mesh approach developed in the research predicts multi-physics, its induced porosity, and the cumulative effect on the residual stress in the powder bed fusion processed Ti-6Al-4V thin wall structures. The results of the residual stress in the multi-layered component from this method were further quantitatively compared with the non-conforming finite element method. The results show the conforming mesh approach was not only effective in capturing the layer geometry, and defects induced during the printing, but also predicted the residual stress in the region of the defect more accurately than the non-conforming mesh methods.

A nonconforming surface mesh generation method by binary tree

Computer-aided engineering (CAE) has emerged as an indispensable tool for facilitating engineering practices and driving industrial innovation. However, the insufficient quality and efficiency of discretizing complex computer-aided design (CAD) models significantly impede the advancement of CAE calculation accuracy and automation. The presence of “dirty” geometry leads to the fact that it is almost impossible to generate a traditional conforming mesh without geometry repair. In most instances, automating geometry repair proves to be even more challenging than meshing. To cope with intricate CAD model with “dirty” geometry, a novel binary tree surface subdivision method (BSSM) is proposed for automatically generated conforming and nonconforming meshes directly on CAD models. In contrast to the conventional conforming mesh, the nonconforming mesh allows for hanging nodes, thereby eliminating the limitations of the mesh conformance. This facilitates rapid transitions in mesh size for automatic mesh generation when dealing with “dirty” geometry. A series of numerical models employing BSSM are presented in this study. Results reveal that BSSM can automatically, efficiently and reliably generate high level quality mesh for arbitrary structures.

A generalization of B-differentiable equations method for elastoplastic contact problems with dual mortar discretization

In this study, a novel method based on the B-differentiable equations (BDEs) is proposed to solve the three-dimensional (3D) static and dynamic elastoplastic contact problems with friction. The contact conditions are formulated as the BDEs system, which can precisely satisfy Coulomb’s law of friction. Based on the mortar segment-to-segment contact model, the contact constraints are enforced by dual Lagrange multipliers for nonconforming meshes of the contact boundary. The non-smooth and nonlinear system equations composed of equilibrium and contact equations are simultaneously solved by the B-differentiable damping Newton method (BDNM), which has global convergence property. In each iteration, firstly, some contact displacements and contact forces, called substituted contact variables, are expressed by others according to the different contact states and contact conditions. The equilibrium equation is then solved, where the stiffness matrix is updated by substituting the contact variables. At last, the substituted contact variables will be restored. In the proposed method, the contact flexibility matrix is no longer needed, which saves the time for calculating the contact flexibility matrix due to changes in the stiffness matrix resulting from the update of stresses in the iteration step. By comparing with the results obtained by finite element commercial software ANSYS, the accuracy of the proposed algorithm is verified, and the property of convergence and efficiency are demonstrated for the elastoplastic contact problems by several numerical examples.

Statistical analysis of effective crack properties by microstructure reconstruction and phase-field modeling

Understanding the relation between the microstructure and the material’s effective behavior is an important aspect in inverse computational materials engineering. Especially in the context of localized, inelastic phenomena like plasticity and crack growth, the microstructure morphology plays a crucial role. Due to the stochastic nature of heterogeneous media, a statistical analysis over multiple simulations is necessary, since even with the same material, the simulated crack paths and effective crack lengths are highly dependent on the specific locations of microstructural features. A relevant factor that limits this type of investigation is the high cost of real microstructure data. This work presents a digital workflow for exploring the fracture properties of materials. Therein, the required statistical analyses are facilitated by an algorithm that reconstructs multiple realization of a material structure given a single example. The reconstructed structures are discretized with a regular non-conforming mesh with a diffuse interface and crack representation. Crack phase-field simulations are conducted in order to analyze the effective response. An in-depth introduction to the required methods is given together with a statistical evaluation of the conducted numerical experiments. It is concluded that the statistical variation of the effective material behavior overshadows morphological trends in the presented case. This confirms the relevance and utility of complementing simulation-based workflows with microstructure reconstruction and statistical analyses.

Symmetric Formulation for Non-Conforming Meshes in Magnetostatic Problems

Symmetric Formulation for Non-Conforming Meshes in Magnetostatic Problems

Linear Displacement Boundary Condition Replacing Periodic Boundary Condition to Easily and Efficiently Homogenize the Elastic Properties of Composites

Periodic boundary condition is commonly used in the FE homogenization method for predicting the thermo-mechanical properties of composites. However, its existing numerical algorithms are generally inefficient. Linear displacement boundary condition is much easier and more efficient to implement for the Representative Volume Elements (RVEs) with conformal and nonconformal surface meshes. Aiming at replacing the periodic boundary condition by the linear displacement boundary condition to easily and efficiently predict the thermo-mechanical properties of composites, this paper compares the implementation efficiency and prediction accuracy of the periodic and linear displacement boundary conditions. Regarding the prediction of the elastic properties of composites, the balance of micro-energy and macro-energy of the RVEs imposed the periodic and linear displacement boundary conditions is checked, and the results demonstrate that these two boundary conditions guarantee the Hill lemma (i.e. the balance of micro-energy and macro-energy of the RVEs). Moreover, comparing with the experimental test and the multi-step micro-mechanical method, the effectiveness of the periodic and linear displacement boundary conditions to predict the elastic properties of composites is validated.

Energy Exascale Computational Fluid Dynamics Simulations With the Spectral Element Method

Abstract Development and application of the open-source GPU-based fluid-thermal simulation code, NekRS, are described. Time advancement is based on an efficient kth-order accurate timesplit formulation coupled with scalable iterative solvers. Spatial discretization is based on the high-order spectral element method (SEM), which affords the use of fast, low-memory, matrix-free operator evaluation. Recent developments include support for nonconforming meshes using overset grids and for GPU-based Lagrangian particle tracking. Results of large-eddy simulations of atmospheric boundary layers for wind-energy applications as well as extensive nuclear energy applications are presented.

The boundary element method for acoustic transmission with nonconforming grids

Acoustic wave propagation through a homogeneous material embedded in an unbounded medium can be formulated as a boundary integral equation and accurately solved with the boundary element method. The computational efficiency deteriorates at high frequencies due to the increase in mesh size with a fixed number of elements per wavelength and also at high material contrasts due to the ill-conditioning of the linear system. This study presents the design of boundary element methods feasible for nonconforming surface meshes at the material interface. The nonconforming algorithm allows for independent grid generation, improves flexibility, and reduces the degrees of freedom. It works for different boundary integral formulations for Helmholtz transmission problems, operator preconditioning, and coupling with finite element solvers. The extensive numerical benchmarks at canonical configurations and an acoustic foam model confirm the significant improvements in computational efficiency when employing the nonconforming grid coupling in the boundary element method.

A FETI B-differentiable equation method for elastic frictional contact problem with nonconforming mesh

A FETI B-differentiable equation method for elastic frictional contact problem with nonconforming mesh

Extended quasicontinuum methodology for highly heterogeneous discrete systems

SummaryLattice networks are indispensable to study heterogeneous materials such as concrete or rock as well as textiles and woven fabrics. Due to the discrete character of lattices, they quickly become computationally intensive. The QuasiContinuum (QC) Method resolves this challenge by interpolating the displacement of the underlying lattice with a coarser finite element mesh and sampling strategies to accelerate the assembly of the resulting system of governing equations. In lattices with complex heterogeneous microstructures with a high number of randomly shaped inclusions the QC leads to an almost fully‐resolved system due to the many interfaces. In the present study the QC Method is expanded with enrichment strategies from the eXtended Finite Element Method (XFEM) to resolve material interfaces using nonconforming meshes. The goal of this contribution is to bridge this gap and improve the computational efficiency of the method. To this end, four different enrichment strategies are compared in terms of their accuracy and convergence behavior. These include the Heaviside, absolute value, modified absolute value and the corrected XFEM enrichment. It is shown that the Heaviside enrichment is the most accurate and straightforward to implement. A first‐order interaction based summation rule is applied and adapted for the extended QC for elements intersected by a material interface to complement the Heaviside enrichment. The developed methodology is demonstrated by three numerical examples in comparison with the standard QC and the full solution. The extended QC is also able to predict the results with 5% error compared to the full solution, while employing almost one order of magnitude fewer degrees of freedom than the standard QC and even more compared to the fully‐resolved system.

A discrete fracture-matrix approach based on Petrov-Galerkin immersed finite element for fractured porous media flow on nonconforming mesh

In this paper, the Petrov-Galerkin immersed finite element method is developed for simulating flow in fractured porous media. We consider the hybrid-dimensional fracture model as an elliptic interface problem with multiple physical coupling interface conditions, and embed the information of low-dimensional fractures into the local functions of interface elements. The immersed finite element space is constructed by explicitly representing the local solution functions in the interface element. This approach addresses the limitation of conforming meshes. More importantly, it can capture both the continuity and discontinuity of pressure, making it effective for both conductive and blocking fractures. We provide a detailed illustration of how to construct local functions on triangular and rectangular meshes. Finally, numerical examples and benchmarks are presented to validate the effectiveness of our proposed method.

Analyzing free vibration and buckling of heated laminated plate with cutouts: A Nitsche-based isogeometric approach

In this research, a comprehensive Nitsche-based isogeometric analysis framework is presented for studying the free vibration and buckling characteristics of laminated plates with cutouts under uniform and non-uniform thermal loads. The approach utilizes non-overlapping non-uniform rational B-Splines patches to discretize the perforated plate and seamlessly connects them using the Nitsche method. The technique is then applied to investigate the free vibration and buckling behaviors of laminated plates with cutouts under varying thermal conditions. Three tests with different element sizes are used to analyze composite plate under uniform thermal load, demonstrating good convergence and accuracy of the present method. Rectangular laminated plates without cutouts, considering various boundary conditions, fiber orientations, and temperature variations under uniform or non-uniform thermal loads are analyzed. The natural frequencies and critical buckling temperatures are found to be in excellent agreement with existing literature. Additionally, the analysis is expanded for laminated plates with cutouts under diverse boundary conditions, considering variations in fiber orientations, cutout shapes and numbers, and types of thermal loads. Thorough investigations are carried out to understand the influence of these factors on the vibrational and buckling behaviors of the plates. Moreover, numerical study with non-conforming mesh demonstrates strong capability of the proposed method.

A comparative study of interpolation algorithms on non-matching meshes for PFEM-FEM fluid-structure interactions

In simulations involving fluid-structure interactions, the partitioned coupling approach allows calculating the fluid flow and the structural displacement using different solvers, which are coupled by a structure-to-fluid algorithm. This partitioned scheme can result in non-conforming meshes at the solid-fluid interface, especially when the fluid and the structure meshes contain elements of different characteristic sizes. In this context, a mesh-interpolation technique is required to transmit the nodal information from one mesh to the other. This paper focuses on comparing radial basis functions and k-nearest neighbours interpolations on fluid-structure interaction test cases. Indeed, those are fast and flexible techniques requiring no topological information other than relative distances between nodes, allowing a straightforward transfer of nodal data between non-conforming meshes. This feature is particularly favourable in the Particle Finite Element Method (PFEM), where remeshing procedures result in significant changes of the finite element discretisation at the fluid-structure interface. Accordingly, our contribution compares the interpolations in a PFEM/FEM partitioned scheme, using 2D and 3D benchmark problems.

Encapsulated generalized summation-by-parts formulations for curvilinear and non-conforming meshes

We extend the construction of so-called encapsulated global summation-by-parts operators to the general case of a mesh which is not boundary conforming. Owing to this development, energy stable discretizations of nonlinear and variable coefficient initial boundary value problems can be formulated in simple and straightforward ways using high-order accurate operators of generalized summation-by-parts type. Encapsulated features on a single computational block or element may include polynomial bases, tensor products as well as curvilinear coordinate transformations. Moreover, through the use of inner product preserving interpolation or projection, the global summation-by-parts property is extended to arbitrary multi-block or multi-element meshes with non-conforming nodal interfaces.

A Multi-Resolution Preconditioner for Nonconformal Meshes in the MoM Solution of Large Multiscale Structures

The paper presents a multi-resolution preconditioner able to improve the solution convergence, via the method of moments and the multilevel fast multipole algorithm, in the case of non-conformal meshes applying the multi-branch Rao-Wilton-Glisson basis functions. The proposed preconditioner enables, for the first time, an automatic multi-level quasi-Helmholtz decomposition on non-conforming meshes, including also the generation of the topological (global) loop functions. Moreover, the generation of the proposed preconditioning scheme is fully parallelized in a multicore shared-memory enviroment. Numerical results show the great flexibility of this approach for the solution of electrically-large multi-scale objects including h-refinement discretizations.

Mortar Method for 2D Elastic Bounded Contact Problems

Abstract This paper presents a contribution to the field of numerical solutions for contact problems, which pose significant challenges in engineering and simulations. Specifically, we address the intricate task of connecting bodies that have been discretized using non-conforming and non-overlapping meshes. Our primary focus lies in investigating the efficacy of the mortar method with a segment-to-segment approach. In this context, we provide a concise overview of the underlying theoretical framework and present our implementation in the MATLAB programming environment. To ascertain the reliability and accuracy of our proposed methodology, we conduct a rigorous validation study by comparing the outcomes obtained from our implementation with those derived from the widely adopted commercial software, ANSYS. To enable a comprehensive evaluation, we select specific benchmark problems that involve the interaction of two elastic bodies. Through a meticulous analysis and comparison of results, we demonstrate the effectiveness and robustness of our approach. The findings of this study contribute substantively to the advancement of numerical techniques for solving contact problems. The validated methodology not only establishes a solid foundation for future research endeavors but also offers a reliable framework for conducting simulations in this domain. Furthermore, the insights gained from this study can potentially facilitate the development of more efficient and accurate computational algorithms for addressing contact problems encountered in various engineering applications.

A Numerical Test Rig for Turbomachinery Flows Based on Large Eddy Simulations With a High-Order Discontinuous Galerkin Scheme—Part I: Sliding Interfaces and Unsteady Row Interactions

Abstract In this first paper of a three-part series, we present the extension and validation of the high-order discontinuous Galerkin scheme in DLR’s CFD-solver trace for scale-resolving simulations of unsteady row interactions. The translational movement of rows in linear cascade experiments is represented in the numerical model by solving the equations in the relative frame of reference. To couple rows in different frames of reference, a sliding interface approach based on the mortar technique for non-conforming meshes has been developed. The verification of the approach is exemplified by three canonical test cases. First, the experimental order of convergence is verified for the isentropic vortex convection. Subsequently, the suitability of the sliding interface approach for scale-resolving simulations is tested on the Taylor–Green vortex flow and a turbulent cylinder flow. Finally, the LES solver is applied to the T106D cascade with upstream moving bars at an exit Reynolds number of 200,000 and exit Mach number of 0.4. The flow physics with and without bars is discussed in terms of the instantaneous flow field, and time- and phase-averaged quantities. The comparison with experimental data shows overall a good agreement, especially for the total pressure losses in the wake, but also reveals uncertainties related to the reproduction of an experiment in the numerical model.

Lamination-based efficient treatment of weak discontinuities for non-conforming finite element meshes

When modelling discontinuities (interfaces) using the finite element method, the standard approach is to use a conforming finite-element mesh in which the mesh matches the interfaces. However, this natural approach can prove cumbersome if the geometry is complex, in particular in 3D. In this work, we develop an efficient technique for a non-conforming finite-element treatment of weak discontinuities by using laminated microstructures. The approach is inspired by the so-called composite voxel technique that has been developed for FFT-based spectral solvers in computational homogenization. The idea behind the method is rather simple. Each finite element that is cut by an interface is treated as a simple laminate with the volume fraction of the phases and the lamination orientation determined in terms of the actual geometrical arrangement of the interface within the element. The approach is illustrated by several computational examples relevant to the micromechanics of heterogeneous materials. Elastic and elastic-plastic materials at small and finite strain are considered in the examples. The performance of the proposed method is compared to two alternative, simple methods showing that the new approach is in most cases superior to them while maintaining the simplicity.

A parallel geometric multigrid method for adaptive topology optimization

This work presents an efficient parallel geometric multigrid (GMG) implementation for preconditioning Krylov subspace methods solving differential equations using non-conforming meshes for discretization. The approach does not constrain such meshes to the typical multiscale grids used by Cartesian hierarchical grid methods, such as octree-based approaches. It calculates the restriction and interpolation operators for grid transferring between the non-conforming hierarchical meshes of the cycle scheme. Using non-Cartesian grids in topology optimization, we reduce the mesh size discretizing only the design domain and keeping the geometry of boundaries in the final design. We validate the GMG method operating on non-conforming meshes using an adaptive density-based topology optimization method, which coarsens the finite elements dynamically following a weak material estimation criterion. The GMG method requires the generation of the hierarchical non-conforming meshes dynamically from the one used by the adaptive topology optimization to analyze to the one coarsening all the mesh elements until the coarsest level of the mesh hierarchy. We evaluate the performance of the adaptive topology optimization using the GMG preconditioner operating on non-conforming meshes using topology optimization on a fine-conforming mesh as the reference. We also test the strong and weak scaling of the parallel GMG preconditioner with two three-dimensional topology optimization problems using adaptivity, showing the computational advantages of the proposed method.