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Open research questions in Numerical methods in engineering

177 gap statements mined from Numerical methods in engineering papers in our 4.5M-paper local library, which holds 2,925 papers on the topic — drawn mostly from each paper's own stated research gap, future-work, challenge and limitation notes, and its abstract. The ones listed below are a selection still marked open; each names the study that raised it, with a DOI link where the paper has one.

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What the literature leaves open

  • The need to capture sharp cracks and accurate crack propagation, kinking, and branching. The challenge of training C-FNO models on limited FE data.

    Parametric phase-field brittle fracture computations with convolutional Fourier neural operators · 2026 · DOI

  • The lack of a comprehensive modeling approach for explosive responses. The limitations of prior work in modeling complex explosive phenomena. The need for effective coupling between solid and gas phases.

    Novel coupled Lagrangian–Eulerian peridynamics: a unified framework for modeling explosive responses with solid and gas interactions · 2026 · DOI

  • The need to ensure that the element size is sufficiently small to accurately resolve the phase-field profile. The challenge of capturing complex crack geometries and topologies in phase-field fracture models.

    A correction method for crack area overestimation in phase-field fracture · 2026 · DOI

  • The need for a modular and flexible method for analyzing and simulating quadcopter drones. The lack of a straightforward design framework for estimating stress contributions from microscale and macroscale. The importance of considering the load-sharing between phases at the microstructural scale in drone design.

    Modular Voxel-DEM based design and analysis of quadcopter drones · 2026 · DOI

  • The study does not provide a comprehensive comparison with other meshless methods. The proposed method may still struggle to accurately capture local features in complex domains. The study is limited to numerical experiments and does not include experimental validation.

    A neural network-augmented patch adaptive meshless method for 3D solid mechanics · 2026 · DOI

  • Existing methods for dynamic fracture simulation have limitations, such as struggling to resolve stress singularities at crack tips. The paper identifies a need for a practical finite element approach that can accurately simulate dynamic crack propagation.

    A practical finite element approach for simulating dynamic crack growth in a proof-of-concept interconnect structure · 2026 · DOI

  • The gap in the literature is the lack of a systematic and quantitative assessment of the influence of different mapping techniques on the accuracy of PFEM simulations. The study identifies the need for a reliable, accurate, and computationally efficient choice for PFEM simulations in geotechnical engineering.

    Impact of mapping techniques and element formulation on PFEM simulations · 2026 · DOI

  • The lack of a robust and efficient method to guarantee global convergence of Newton's method in phase-field brittle fracture computations. The need for a reliable and accurate approach to model the nucleation and propagation of cracks in brittle materials.

    Iterative convergence in phase-field brittle fracture computations: exact line search is all you need · 2026 · DOI

  • There is a need for a peridynamic model to simulate the behavior of smart beams under thermomechanical loading. Prior work has not fully addressed the simulation of SMA materials using peridynamic modeling.

    Peridynamic modeling of shape memory alloy smart beam under thermomechanical loading · 2026 · DOI

  • The standard Periodic BCs do not account for the orientation of the localization band. The Percolation-path-aligned BCs suffer from several drawbacks, including uneven straining of the weakly and strongly constrained sections. There is a need for novel boundary conditions that can accurately capture the development of strain localization in softening heterogeneous materials.

    On Representation of Macroscopic Crack in Periodic Fine‐Scale Discrete Mechanical Models · 2026 · DOI

  • The study only investigated the influence of time resolution on the flow field. The study only used a limited number of test cases. The study did not compare the results with experimental data.

    A pressure-projection formulation in a least-squares meshfree method for the incompressible Navier–Stokes equations using a staggered-variable arrangement · 2026 · DOI

  • Limited observability of real-world systems in geomechanical modeling. The need for frequent remeshing and computational impracticality in meshfree methods. The lack of differentiability in traditional simulation pipelines.

    JAX-MPM: a learning-augmented differentiable meshfree framework for GPU-accelerated Lagrangian simulation and geophysical inverse modeling · 2026 · DOI

  • Existing methods are constrained to problems with simple geometries and boundary conditions. Lack of a robust and accurate numerical model for simulating anisotropic heat conduction in phase-change problems. Need for a model that can handle complex geometries and boundary conditions.

    A peridynamics-based anisotropic heat conduction model for phase-change problems · 2026 · DOI

  • Evaluating the fracture surface energy dissipated over the crack area is intractable due to the surface changes that occur during crack growth. The BVP problem is ill-posed due to the lack of boundary conditions.

    Phase-field fracture modeling in porous materials with idealized and realistic morphologies · 2026 · DOI

  • The bond-based PD model has limitations, such as restricted Poisson's ratio and difficulty in modeling anisotropic properties. Existing numerical techniques have limitations in capturing crack propagation. There is a need for a new model that can unify isotropy and anisotropy under the same theoretical framework.

    An area-based peridynamic theory for failure analysis · 2026 · DOI

  • Limited capability of existing finite element formulations in handling large displacements and contact interactions.

    An improved edge-based smoothed finite element formulation for nonlinear analysis of triangular shell elements and its application · 2026 · DOI

  • Investigating the effect of voids on stress concentration in different materials, - Analyzing the impact of void size and shape on stress singularities, - Examining the behavior of stress singularities in 3D…

    Analysis of stress singularities induced by a void at bi-material interfaces under tensile loading using FEM · 2026 · DOI

  • The study remains in the realm of the most widely used model to date, - The initial trainings were conducted with grids of size 128 × 128, which resulted in too diffused results, - The study only considers two-dimensional SEN…

    Parametric phase-field brittle fracture computations with convolutional Fourier neural operators · 2026 · DOI

  • The Element-Free Galerkin (EFG) method requires a background mesh, - EFG has no intrinsic fracture (needs XEFG/damage model), - Essential BCs are non-trivial for EFG, - SPG is computationally more expensive than SPH and sensitive to parameters…

    Comparative numerical assessment of SPH, EFG, and SPG methods for orthogonal cutting of steel · 2026 · DOI

  • Further studies are needed to investigate the applicability of SPG to other machining processes, - Research is required to improve the computational efficiency of SPG, - Investigations into the effects of different material models and parameters on the performance of SPG…

    Comparative numerical assessment of SPH, EFG, and SPG methods for orthogonal cutting of steel · 2026 · DOI

  • Further in-depth investigation of explosive problems across diverse scenarios, - Research on the neighborhood information of material points at each time step, - Development of more efficient algorithms for simulating explosion…

    Novel coupled Lagrangian–Eulerian peridynamics: a unified framework for modeling explosive responses with solid and gas interactions · 2026 · DOI

  • Further investigation of the proposed method for more complex flow problems. Comparison of the proposed method with other numerical methods. Extension of the proposed method to three-dimensional flow problems.

    A pressure-projection formulation in a least-squares meshfree method for the incompressible Navier–Stokes equations using a staggered-variable arrangement · 2026 · DOI

  • Exploring differentiable programming as a flexible and scalable alternative for inverse modeling in geophysical hazards. Developing meshfree methods to improve robustness in large-deformation problems. Investigating the application of JAX-MPM to other geomechanical simulations.

    JAX-MPM: a learning-augmented differentiable meshfree framework for GPU-accelerated Lagrangian simulation and geophysical inverse modeling · 2026 · DOI

  • Continuous re-inspection of structures and parts during design life is impractical and difficult, - The BVP problem is ill-posed due to the lack of boundary conditions, - Microscale heterogeneity may present as both constitutive anisotropy and a direction dependent critical energy release…

    Phase-field fracture modeling in porous materials with idealized and realistic morphologies · 2026 · DOI

  • Further research can be done to apply the ABPD model to other materials and structures, - The ABPD model can be used to study the behavior of materials under different loading…

    An area-based peridynamic theory for failure analysis · 2026 · DOI

Most-cited papers in Numerical methods in engineering

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Open questions raised by papers listed on this page include: (1) The lack of a comprehensive modeling approach for explosive responses (“Novel coupled Lagrangian–Eulerian peridynamics: a unified framework for…”, 2026); (2) The lack of a straightforward design framework for estimating stress contributions from microscale and macroscale (“Modular Voxel-DEM based design and analysis of quadcopter drones”, 2026); (3) The gap in the literature is the lack of a systematic and quantitative assessment of the influence of different mapping techniques on the accuracy of PFEM simulations (“Impact of mapping techniques and element formulation on PFEM simulations”, 2026); (4) The lack of a robust and efficient method to guarantee global convergence of Newton's method in phase-field brittle fracture computations (“Iterative convergence in phase-field brittle fracture computations: exact line…”, 2026); (5) There is a need for a peridynamic model to simulate the behavior of smart beams under thermomechanical loading (“Peridynamic modeling of shape memory alloy smart beam under thermomechanical…”, 2026). 177 gap statements have been mined from Numerical methods in engineering papers in our 4.5M-paper local library, which holds 2,925 papers on the topic; the gaps come from whichever of those papers state one. They are mostly the research gaps the authors state and the papers' abstracts, plus future-work, limitations and challenges passages. The ones listed below are a selection still marked open; each names the study that raised it, with a DOI link where the paper has one, so you can read the original claim in context.

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