High-order Finite Element Method Research Articles

A high-order multiscale finite element method for 3D direct current resistivity log modeling

When solving the 3D direct current resistivity logging problem with complex heterogeneous structure, 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 lead to an excessively large calculation, thus occupying a large amount of computing resources and making it difficult to solve. The multiscale finite element method can significantly reduce the size of the coefficient matrix during the solving process, which greatly reduces the computational cost. Nevertheless, the low-order multiscale finite element cannot accurately transfer the heterogeneous information of the fine mesh to the coarse mesh, it leads to numerical errors. And the imposition of linear boundary conditions to construct multiscale basis functions may introduce resonance error. Consequently, this article implements a 3D direct current resistivity logging forward algorithm based on high-order multiscale finite element. On the one hand, we construct boundary and internal high-order multiscale basis functions by utilizing arbitrary order orthogonal polynomials within local cells, and then linearly combine them to form complete high-order multiscale basis functions, this ensures the approximation ability and convergence accuracy of the numerical solution. On the other hand, we construct temporary high-order multiscale basis functions on extended local cells, combined with an oversampling technique, which can mitigate the error problem caused by boundary resonance effects. Simulate in the scenarios with three different electrode configurations: pole-pole, pole-dipole, and dipole-dipole. The results show that our new method can approximate the fine-scale reference solution on the coarse mesh with high accuracy and significantly reduced computational time at the linear system solving stage.

A High-Order Shifted Interface Method for Lagrangian Shock Hydrodynamics

We present a new method for two-material Lagrangian hydrodynamics, which combines the Shifted Interface Method (SIM) with a high-order Finite Element Method. Our approach relies on an exact (or sharp) material interface representation, that is, it uses the precise location of the material interface. The interface is represented by the zero level-set of a continuous high-order finite element function that moves with the material velocity. This strategy allows to evolve curved material interfaces inside curved elements. By reformulating the original interface problem over a surrogate (approximate) interface, located in proximity of the true interface, the SIM avoids cut cells and the associated problematic issues regarding implementation, numerical stability, and matrix conditioning. Accuracy is maintained by modifying the original interface conditions using Taylor expansions. We demonstrate the performance of the proposed algorithms on established numerical benchmarks in one, two and three dimensions.

Accelerating composite sandwich plate analysis: A hybrid higher-order XFEM and ANN approach for natural frequency prediction

Free vibration analysis of laminated composites plays a critical role in design optimization, material selection, structural integrity assessment, vibration suppression, and fatigue life prediction. A novel coupled higher-order extended finite element method (HO-XFEM) and artificial neural network (ANN) approach is implemented for predicting the natural frequency of sandwich plates with a composite layout of 0°/θ°/0°/core/0°/θ°/0°, where θ varies from 15 to 90 degrees. A total of 557 data points are generated through HO-XFEM, which is then utilized to construct an optimized ANN model for predicting nondimensionalized natural frequency. The ANN model, designed specifically to predict the first mode of natural frequency, is optimized with a structure of 4-90-90-1 and softmax transfer functions in intermediate layers. It predicts the nondimensional natural frequency with the average, minimum, and maximum mean-squared error values of 0.0984 %, 0.00339 %, and 0.49 %, respectively. The optimized ANN model computes the natural frequency for different sandwich plate configurations a few million times faster compared to HO-XFEM and thus establishes a transformative framework for real-time structural integrity assessment.

Strength Analysis and Optimization Recommendations for the Cargo Hold Sections of Ultra-Large Container Ships Based on an Improved High-Order Finite Element Method

Strength Analysis and Optimization Recommendations for the Cargo Hold Sections of Ultra-Large Container Ships Based on an Improved High-Order Finite Element Method

Optimizing structural resilience of composite smart structures using predictive artificial intelligence and Carrera unified formulation: A new approach to improve the efficiency of smart building construction

This research introduces a novel approach for optimizing structural resilience in smart building construction, focusing on enhancing both dynamic performance and energy efficiency. The study investigates a sandwich plate structure, where the core is made of functionally graded (FG)-triply periodic minimal surfaces (TPMS) material, providing superior mechanical properties. The face sheets are equipped with sensors and actuators, forming an active system capable of real-time monitoring and adaptive response to external forces. To further improve the system’s stability, a viscoelastic-auxetic foundation is incorporated, offering enhanced damping characteristics and the ability to absorb vibrations more efficiently. The ultimate goal is to optimize the natural frequency of the sandwich plate to prevent resonance under dynamic loading conditions, ensuring long-term durability, and performance. The structural model is developed using the Carrera unified formulation (CUF), a high-order finite element method that accurately represents complex geometries and deformation behavior, enabling precise simulations of the dynamic response. Artificial intelligence (AI) techniques are then employed to validate the model, optimizing parameters through machine learning algorithms to enhance the efficiency of the structural system. This new methodology combines advanced materials with cutting-edge computational tools to offer a significant improvement in the construction of smart buildings. By optimizing both structural resilience and energy efficiency, this approach provides a pathway to more sustainable and durable building designs, marking a step forward in the future of intelligent, high-performance construction technologies.

Transcranial ultrasound modeling using the spectral-element method.

This work explores techniques for accurately modeling the propagation of ultrasound waves in lossy fluid-solid media, such as within transcranial ultrasound, using the spectral-element method. The objectives of this work are twofold, namely, (1) to present a formulation of the coupled viscoacoustic-viscoelastic wave equation for the spectral-element method in order to incorporate attenuation in both fluid and solid regions and (2) to provide an end-to-end workflow for performing spectral-element simulations in transcranial ultrasound. The matrix-free implementation of this high-order finite-element method is very well-suited for performing waveform-based ultrasound simulations for both transcranial imaging and focused ultrasound treatment thanks to its excellent accuracy, flexibility for dealing with complex geometries, and computational efficiency. The ability to explicitly mesh distinct interfaces between regions with high impedance contrasts eliminates staircasing artifacts, which are otherwise non-trivial to mitigate within discretization approaches based on regular grids. This work demonstrates the efficacy of this modeling technique for transcranial ultrasound through a number of numerical examples. While the examples in this work primarily focus on transcranial applications, this type of modeling is equally relevant within other soft tissue-bone systems such as in limb or spine imaging.

Experience and analysis of scalable high-fidelity computational fluid dynamics on modular supercomputing architectures

The never-ending computational demand from simulations of turbulence makes computational fluid dynamics (CFD) a prime application use case for current and future exascale systems. High-order finite element methods, such as the spectral element method, have been gaining traction as they offer high performance on both multicore CPUs and modern GPU-based accelerators. In this work, we assess how high-fidelity CFD using the spectral element method can exploit the modular supercomputing architecture at scale through domain partitioning, where the computational domain is split between a Booster module powered by GPUs and a Cluster module with conventional CPU nodes. We investigate several different flow cases and computer systems based on the Modular Supercomputing Architecture (MSA). We observe that for our simulations, the communication overhead and load balancing issues incurred by incorporating different computing architectures are seldom worthwhile, especially when I/O is also considered, but when the simulation at hand requires more than the combined global memory on the GPUs, utilizing additional CPUs to increase the available memory can be fruitful. We support our results with a simple performance model to assess when running across modules might be beneficial. As MSA is becoming more widespread and efforts to increase system utilization are growing more important our results give insight into when and how a monolithic application can utilize and spread out to more than one module and obtain a faster time to solution.

A novel flexible infinite element for transient acoustic simulations

This article addresses the efficient solution of exterior acoustic transient problems using the Finite Element Method (FEM) in combination with infinite elements. Infinite elements are a popular technique to enforce non-reflecting boundary conditions. The Astley–Leis formulation presents several advantages in terms of ease of implementation, and results in frequency-independent system matrices, that can be used for transient simulations of wave propagation phenomena. However, for time-domain simulations, the geometrical flexibility of Astley–Leis infinite elements is limited by time-stability requirements. In this article, we present a novel infinite element formulation, called flexible infinite element, for which the accuracy does not depend on the positioning of the virtual sources. From a software implementation perspective, the element proposed can be seen as a specialized FEM element and can be easily integrated into a high-order FEM code. The effectiveness of the flexible formulation is demonstrated with frequency and time-domain examples; for both cases, we show how the flexible infinite elements can be attached to arbitrarily-shaped convex FE boundaries. In particular, we show how the proposed technique can be used in combination with existing model order reduction strategies to run fast and accurate transient simulations.

Mean field control of droplet dynamics with high-order finite-element computations

Liquid droplet dynamics are widely used in biological and engineering applications, which contain complex interfacial instabilities and pattern formation such as droplet merging, splitting and transport. This paper studies a class of mean field control formulations for these droplet dynamics, which can be used to control and manipulate droplets in applications. We first formulate the droplet dynamics as gradient flows of free energies in modified optimal transport metrics with nonlinear mobilities. We then design an optimal control problem for these gradient flows. As an example, a lubrication equation for a thin volatile liquid film laden with an active suspension is developed, with control achieved through its activity field. Lastly, we apply the primal–dual hybrid gradient algorithm with high-order finite-element methods to simulate the proposed mean field control problems. Numerical examples, including droplet formation, bead-up/spreading, transport, and merging/splitting on a two-dimensional spatial domain, demonstrate the effectiveness of the proposed mean field control mechanism.

Mixed finite elements of higher-order in elastoplasticity

In this paper a higher-order mixed finite element method for elastoplasticity with linear kinematic hardening is analyzed. Thereby, the non-differentiability of the involved plasticity functional is resolved by a Lagrange multiplier leading to a three field formulation. The finite element discretization is conforming in the displacement field and the plastic strain but potentially non-conforming in the Lagrange multiplier as its Frobenius norm is only constrained in a certain set of Gauss quadrature points. A discrete inf-sup condition with constant 1 and the well posedness of the discrete mixed problem are shown. Moreover, convergence and guaranteed convergence rates are proved with respect to the mesh size and the polynomial degree, which are optimal for the lowest order case. Numerical experiments underline the theoretical results.

Comparison between a priori and a posteriori slope limiters for high-order finite volume schemes

High-order finite volume and finite element methods offer impressive accuracy and cost efficiency when solving hyperbolic conservation laws with smooth solutions. However, if the solution contains discontinuities, these high-order methods can introduce unphysical oscillations and severe overshoots/undershoots. Slope limiters are an effective remedy, combating these oscillations by preserving monotonicity. Some limiters can even maintain a strict maximum principle in the numerical solution. They can be classified into one of two categories: a priori and a posteriori limiters. The former revises the high-order solution based only on data at the current time tn, while the latter involves computing a candidate solution at tn+1 and iteratively recomputing it until some conditions are satisfied. These two limiting paradigms are available for both finite volume and finite element methods.In this work, we develop a methodology to compare a priori and a posteriori limiters for finite volume solvers at arbitrarily high order. We select the maximum principle preserving scheme presented in [1,2] as our a priori limited scheme. For a posteriori limiting, we adopt the methodology presented in [3] and search for so-called troubled cells in the candidate solution. We revise these cells using a robust MUSCL fallback scheme, diverging from the traditional Multi-dimensional Optimal Order Detection (MOOD) framework by allowing each candidate solution to undergo only a single revision. This modification enables our a posteriori schemes to achieve competitive computational speed while permitting small, though non-zero, maximum principle violations.The linear advection equation is solved in both one and two dimensions and we compare variations of these limited schemes based on their ability to maintain a maximum principle, solution quality over long time integration and computational cost.This analysis reveals a fundamental tradeoff between these three aspects. The high-order a posteriori limited solutions boast excellent quality at long time-scales, taking full advantage of the sharp gradients of the high-order finite volume method. However, their relaxed implementation allows for consistent, albeit small, violations of the maximum principle. In contrast, high-order a priori limited solutions rigorously uphold the maximum principle but suffer from numerical artifacts and diffusion that increasingly dominate at extended time scales. The convex blending of revised fluxes can mitigate the maximum principle violations in a posteriori schemes but introduces more noticeable numerical artifacts. These results suggest a fundamental trade-off between robustness against numerical diffusion and the preservation of a strict maximum principle.Moreover, we compare two methods for computing the flux integrals along two-dimensional cell faces, revealing that one option is more cost-effective but leads to particularly large violations when used with the a priori limited scheme. Consequently, the a priori limited scheme is forced to use the more costly flux computation, making it significantly more expensive at higher-order than the a posteriori limited scheme on CPU architecture. This cost difference can be almost entirely mitigated using a GPU implementation of the same schemes, highlighting that GPUs are well-suited for high-order finite volume stencil operations.

Numerical research of functionally graded magneto-electro-elastic structures under thermal environment via the thermo-mechanical-electro-magnetic enriched finite element method

Abstract In this work, the functionally graded magneto-electro-elastic (FG-MEE) structures were used to study mechanical performance under a thermal environment using the thermo-mechanical-electro-magnetic enriched finite element method (TMEM-EFEM). Through enhancing the traditional solution space of the finite element method (FEM), additional degrees of freedom are introduced without modifying the mesh structure. Therefore, as a class of higher-order finite element methods, field variables with high gradients can be captured without a relatively finer mesh. The accuracy of TMEM-EFEM is improved by using interpolation covers. The feasibility of TMEM-EFEM is proven by numerical examples. The proposed TMEM-EFEM is proven to be accurate under the relatively coarse mesh. Through numerical examples, the high accuracy of TMEM-EFEM is demonstrated. Therefore, TMEM-EFEM can be a good alternative for the statics study of FG-MEE structures.

Enhancing vibroacoustic reduced order models for digital twin development in industrial applications

In recent years, the growing interest towards the Digital Twin has prompted extensive exploration in both academic and industrial domains. To realize this technology, the development of simulation models that seamlessly run parallel to the physical assets is necessary. Model Order Reduction (MOR) plays a pivotal role, significantly easing the computational demands of 3D models while maintaining high fidelity. Krylov-based methods have proven successful in various vibroacoustic applications. However, state-of-the-art contributions employ suboptimal modeling strategies, leading to computationally inefficient offline phases and limited scalability. The linear Finite Element Method (FEM) exhibits lower efficiency than high-order FEM; Astley-Leis infinite elements, used to enforce non-reflecting conditions, constrain models to spherical domains, proving inefficient for elongated structures. In this work, advanced techniques-FEM Adaptive Order (FEMAO) and flexible infinite elements- are employed to alleviate the computational burden of the MOR process. The advantages of combining MOR, FEMAO, and flexible infinite elements are explored through various numerical examples. Specifically, we highlight the efficiency of output-based MOR in handling models with a high number of inputs, showcasing practical applications, especially in electric motor noise analysis. In conclusion, these numerical techniques reduce the computational burden of the offline phase, unlocking the Digital Twin for large-scale industrial models.

Comparison of different elastic strain definitions for largely deformed SEI of chemo-mechanically coupled silicon battery particles

Amorphous silicon is a highly promising anode material for next-generation lithium-ion batteries. Large volume changes of the silicon particle have a critical effect on the surrounding solid-electrolyte interphase (SEI) due to repeated fracture and healing during cycling. Based on a thermodynamically consistent chemo-elasto-plastic continuum model we investigate the stress development inside the particle and the SEI. Using the example of a particle with SEI, we apply a higher order finite element method together with a variable-step, variable-order time integration scheme on a nonlinear system of partial differential equations. Starting from a single silicon particle setting, the surrounding SEI is added in a first step with the typically used elastic Green–St-Venant (GSV) strain definition for a purely elastic deformation. For this type of deformation, the definition of the elastic strain is crucial to get reasonable simulation results. In case of the elastic GSV strain, the simulation aborts. We overcome the simulation failure by using the definition of the logarithmic Hencky strain. However, the particle remains unaffected by the elastic strain definitions in the particle domain. Compared to GSV, plastic deformation with the Hencky strain is straightforward to take into account. For the plastic SEI deformation, a rate-independent and a rate-dependent plastic deformation are newly introduced and numerically compared for three half cycles for the example of a radial symmetric particle.

Multiscale Analysis of Sandwich Beams with Polyurethane Foam Core: A Comparative Study of Finite Element Methods and Radial Point Interpolation Method.

This study presents a comprehensive multiscale analysis of sandwich beams with a polyurethane foam (PUF) core, delivering a numerical comparison between finite element methods (FEMs) and a meshless method: the radial point interpolation method (RPIM). This work aims to combine RPIM with homogenisation techniques for multiscale analysis, being divided in two phases. In the first phase, bulk PUF material was modified by incorporating circular holes to create PUFs with varying volume fractions. Then, using a homogenisation technique coupled with FEM and four versions of RPIM, the homogenised mechanical properties of distinct PUF with different volume fractions were determined. It was observed that RPIM formulations, with higher-order integration schemes, are capable of approximating the solution and field smoothness of high-order FEM formulations. However, seeking a comparable field smoothness represents prohibitive computational costs for RPIM formulations. In a second phase, the obtained homogenised mechanical properties were applied to large-scale sandwich beam problems with homogeneous and approximately functionally graded cores, showing RPIM's capability to closely approximate FEM results. The analysis of stress distributions along the thickness of the beam highlighted RPIM's tendency to yield lower stress values near domain edges, albeit with convergence towards agreement among different formulations. It was found that RPIM formulations with lower nodal connectivity are very efficient, balancing computational cost and accuracy. Overall, this study shows RPIM's viability as an alternative to FEM for addressing practical elasticity applications.

An arbitrarily high order unfitted finite element method for elliptic interface problems with automatic mesh generation, Part II. Piecewise-smooth interfaces

We consider the reliable implementation of an adaptive high-order unfitted finite element method on Cartesian meshes for solving elliptic interface problems with geometrically curved singularities. We extend our previous work on the reliable cell merging algorithm for smooth interfaces to automatically generate the induced mesh for piecewise smooth interfaces. An hp a posteriori error estimate is derived for a new unfitted finite element method whose finite element functions are conforming in each subdomain. Numerical examples illustrate the competitive performance of the method.

Modeling and simulation of chemo-elasto-plastically coupled battery active particles

As an anode material for lithium-ion batteries, amorphous silicon offers a significantly higher energy density than the graphite anodes currently used. Alloying reactions of lithium and silicon, however, induce large deformation and lead to volume changes up to 300%. We formulate a thermodynamically consistent continuum model for the chemo-elasto-plastic diffusion-deformation behavior of amorphous silicon and it’s alloy with lithium based on finite deformations. In this paper, two plasticity theories, i.e. a rate-independent theory with linear isotropic hardening and a rate-dependent one, are formulated to allow the evolution of plastic deformations and reduce occurring stresses. Using modern numerical techniques, such as higher order finite element methods as well as efficient space and time adaptive solution algorithms, the diffusion-deformation behavior resulting from both theories is compared. In order to further increase the computational efficiency, an automatic differentiation scheme is used, allowing for a significant speed up in assembling time as compared to an algorithmic linearization for the global finite element Newton scheme. Both plastic approaches lead to a more heterogeneous concentration distribution and to a change to tensile tangential Cauchy stresses at the particle surface at the end of one charging cycle. Different parameter studies show how an amplification of the plastic deformation is affected. Interestingly, an elliptical particle shows only plastic deformation at the smaller half axis. With the demonstrated efficiency of the applied methods, results after five charging cycles are also discussed and can provide indications for the performance of lithium-ion batteries in long term use.

Collimated beam formation in 3D acoustic sonic crystals

We demonstrate strongly collimated beam formation, at audible frequencies, in a three-dimensional acoustic phononic crystal where the wavelength is commensurate with the crystal elements; the crystal is a seemingly simple rectangular cuboid constructed from closely-spaced spheres, and yet demonstrates rich wave phenomena acting as a canonical three-dimensional metamaterial. We employ theory, numerical simulation and experiments to design and interpret this collimated beam phenomenon and use a crystal consisting of a finite rectangular cuboid array of 4×10×10 polymer spheres 1.38 cm in diameter in air, arranged in a primitive cubic cell with the centre-to-centre spacing of the spheres, i.e. the pitch, as 1.5 cm. Collimation effects are observed in the time domain for chirps with central frequencies at 14.2 kHz and 18 kHz, and we deployed a laser feedback interferometer or Self-Mixing Interferometer – a recently proposed technique to observe complex acoustic fields—that enables experimental visualisation of the pressure field both within the crystal and outside of the crystal. Numerical exploration using a higher-order multi-scale finite element method designed for the rapid and detailed simulation of 3D wave physics further confirms these collimation effects and cross-validates with the experiments. Interpretation follows using High Frequency Homogenization and Bloch analysis whereby the different origin of the collimation at these two frequencies is revealed by markedly different isofrequency surfaces of the sonic crystal.

Comparison of Physics Informed Neural Networks and Finite Element Method Solvers for advection-dominated diffusion problems

We present a comparison of Physics Informed Neural Networks (PINN) and Variational Physics Informed Neural Networks (VPINN) with higher-order and continuity Finite Element Method (FEM). We focus on the one-dimensional advection-dominated diffusion problem and the two-dimensional Eriksson–Johnson model problem. We show that the standard Galerkin method for FEM cannot solve this problem on uniform grid. We discuss the stabilization of the advection-dominated diffusion problem with the Petrov–Galerkin (PG) formulation and present the FEM solution obtained with the PG method. The main benefit of using a stabilization method is that it can deliver a good-quality approximation to the solution on a mesh that is not fully refined towards the singularity. We employ PINN and VPINN methods, defining several strong and weak loss functions. We compare the training and solutions of PINN and VPINN methods with higher-order FEM methods. We consider a case with uniform FEM and uniform distribution of points for PINN, as well as uniform distribution of test functions for VPINN. We also consider adaptive FEM, refined towards edge singularity, and non-uniform distribution of points for PINN, as well as non-uniform distribution of test functions for VPINN.

A computational approach to integrate three-dimensional peridynamics and two-dimensional higher-order classical elasticity theory for fracture analysis

AbstractAs a nonlocal alternative of classical continuum theory, peridynamics (PD) is mathematically compatible to discontinuities, making it particularly attractive for failure prediction. The PD theory on the other side can be computationally demanding due to its nonlocal interactions. A coupling between PD and refined higher-order finite element method (FEM) integrates their salient features. The present study proposes a computational approach to couple three-dimensional peridynamics with two-dimensional higher-order finite elements based on classical elasticity. The bond-based PD modeling is considered in a region where damage might appear while refined finite element modeling is used for the remaining region. The refined finite elements employed in this study are based on the 2D Carrera Unified Formulation (CUF), which provides 3D-like accuracy with optimized computational efficiency. The coupling between PD and FEM is achieved through the Lagrange multiplier method which permits physical consistency and compatibility at the interface domain. An adaptive convergence check algorithm is also proposed to achieve predetermined accuracy in the solution with minimum computational effort. Simulations of quasi-static tension tests, wedge splitting tests and L-plate cracking tests are carried out for verification. In-depth analysis shows that the present approach can reproduce the linear deformation, material degradation and crack propagation in an effective way.