Conformal Mesh Research Articles

A hybrid finite element method for moving-habitat models in two spatial dimensions

Moving-habitat models track the density of a population whose suitable habitat shifts as a consequence of climate change. Whereas most previous studies in this area consider 1-dimensional space, we derive and study a spatially 2-dimensional moving-habitat model via reaction-diffusion equations. The population inhabits the whole space. The suitable habitat is a bounded region where population growth is positive; the unbounded complement of its closure is unsuitable with negative growth. The interface between the two habitat types moves, depicting the movement of the suitable habitat poleward. Detailed modelling of individual movement behaviour induces a nonstandard discontinuity in the density across the interface. For the corresponding semi-discretised system we prove well-posedness for a constant shifting velocity before constructing an implicit-explicit hybrid finite element method. In this method, a Lagrange multiplier weakly imposes the jump discontinuity across the interface. For a stationary interface, we derive optimal a priori error estimates over a conformal mesh with nonconformal discretisation. We demonstrate with numerical convergence tests that these results hold for the moving interface. Finally, we demonstrate the strength of our hybrid finite element method with two biologically motivated cases, one for a domain with a curved boundary and the other for non-constant shifting velocity.

Level-set topology optimization with PDE generated conformal meshes

This paper presents a level-set topology optimization approach that uses conformal meshes for the analysis of the displacement field. The structure’s boundary is represented by the iso-contour of a level-set field discretized on a fixed background design mesh. The conformal mesh is updated for each design iteration via a PDE based mesh morphing process that identifies the set of facets in the background mesh that are homeomorphic to the boundary and relaxes the homeomorphic mesh to conform to the structure’s boundary and ensure high element quality. The conformal mesh allows for a more accurate computation of the response versus density and some level-set based methods which interpolate material properties using the volume fraction. Numerical examples illustrate the proposed approach by optimizing linear-elastic two- and three-dimensional structures, wherein insight into the performance of the mesh morphing process is provided. The examples also highlight the scalability of the approach.

Shape Optimization for Lithium-Ion Battery with Porous Electrodes

In this presentation, we introduce a gradient-based shape optimization framework to design porous electrodes for maximum energy storage. Lithium-ion batteries are becoming an increasingly important energy source and storage device. Large surface area that advocates high reaction rates often leads to small porosities that deteriorate electrochemical transport, which significantly restricts ion penetration depth, limiting the material utilization in planar electrodes. Consequently, architected electrodes are required to optimize performance.We consider a model with two materials, namely porous electrode and pure electrolyte. Density-based topology optimization with continuous material volume fraction has previously been implemented on lithium-ion battery system for both half-cell and full-cell model to maximizes its energy storage. Despite the capability of producing complex interdigitated structures, an appropriate penalization scheme for intermediate materials is required for the optimizer to provide near binary designs. Setting up such a penalization is often time-consuming due to the large number of design-dependent variables that affect the forward model differently. Alternatively, the shape optimization approach used in this work produces binary designs naturally by morphing conformal meshes, circumventing inaccuracies associated with fuzzy interfaces. Shape sensitivities are computed via the adjoint method and leveraging automatic differentiation. The optimization problem is solved using a nonlinear programming technique.In this presentation, we perform shape optimization on a half-cell model to maximize its energy storage when a fixed charging current is applied. We study multiple arrangements of physical parameters and compare their performance against a monolithic electrode. Both 2D and 3D results are presented.

Free-form and multi-physical metamaterials with forward conformality-assisted tracing.

Transformation theory, active control and inverse design have been mainstream in creating free-form metamaterials. However, existing frameworks cannot simultaneously satisfy the requirements of isotropic, passive and forward design. Here we propose a forward conformality-assisted tracing method to address the geometric and single-physical-field constraints of conformal transformation. Using a conformal mesh composed of orthogonal streamlines and isotherms (or isothermal surfaces), this method quasi-analytically produces free-form metamaterials using only isotropic media. The geometric nature of this approach allows for universal regulation of both dissipative thermal fields and non-dissipative electromagnetic fields. We experimentally demonstrate free-form thermal cloaking in both two and three dimensions. Additionally, the multi-physical functionalities of our method, including optical cloaking, bending and thermo-electric transparency, confirm its broad applicability. Our method features improvements in efficiency, accuracy and adaptability over previous approaches. This study provides an effective method for designing complex metamaterials with arbitrary shapes across various physical domains.

Multiscale simulation of debonding in fiber-reinforced composites by a combination of MsFEM and XFEM

In this study, a novel multiscale computational method is developed to simulate the nonlinear debonding process in long Fiber Reinforced Composites (FRC). The extended finite element method (XFEM) combined with a cohesive zone model is used to avoid conformal meshing, the perfect fiber-to-matrix bonding is considered as a bi-material interface, and wherever debonding occured, a cohesive crack segment is introduced. Multiscale base functions are constructed numerically to effectively deliver the variables between macroscopic and microscopic. The debonding process in composites with randomly distributed fibers is then numerically investigated, and effective constitutive relation is evaluated on an RVE. The results obtained by the present method are compared with single-scale XFEM and analytical results. It is shown that, the proposed method provides a nonlinear response for the FRC at a considerably reduced computational cost.

Ice Shape Convergence in Multistep Ice Accretion Simulations

Local errors in the geometrical description of the ice front are amplified in multistep simulations over conformal meshes due to the coupling of aerodynamics, water impingement, ice accretion, and grid deformation. Small perturbations in the initial phase of ice formation possibly result in dramatically different ice shapes, which can hinder the stability of the multistep procedure. This problem is analyzed by investigating the combined effects of space and time discretization on the ice growth over airfoils and three-dimensional wings. We propose an automatic procedure for selecting the time interval for the update of the aerodynamics and particle impingement. A local growth limiter λ is introduced here to bound the local ice thickness growth to be comparable to the local grid spacing on the surface, resulting in an automatic adaptive time step to be used in the multistep simulation. Examples are provided for three-dimensional cases under both rime and glaze conditions over straight and swept wings. These examples highlight the different accretion mechanisms of the two ice regimes and preliminarily indicate that ice-shape convergence can be achieved for low values of λ.

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.

Robust Coarse Cage Construction With Small Approximation Errors.

We propose a robust and automatic method to construct manifold cages for 3D triangular meshes. The cage contains hundreds of triangles to tightly enclose the input mesh without self-intersections. To generate such cages, our algorithm consists of two phases: (1) construct manifold cages satisfying the tightness, enclosing, and intersection-free requirements and (2) reduce mesh complexities and approximation errors without violating the enclosing and intersection-free requirements. To theoretically make the first stage have those properties, we combine the conformal tetrahedral meshing and tetrahedral mesh subdivision. The second step is a constrained remeshing process using explicit checks to ensure that the enclosing and intersection-free constraints are always satisfied. Both phases use a hybrid coordinate representation, i.e., rational numbers and floating point numbers, combined with exact arithmetic and floating point filtering techniques to guarantee the robustness of geometric predicates with a favorable speed. We extensively test our method on a data set of over 8500 models, demonstrating robustness and performance. Compared to other state-of-the-art methods, our method possesses much stronger robustness.

A treatment of particle–electrolyte sharp interface fracture in solid-state batteries with multi-field discontinuities

In this work, we present a computational framework for coupled electro-chemo-(nonlinear) mechanics at the particle scale for solid-state batteries. The framework accounts for interfacial fracture between the active particles and solid electrolyte due to intercalation stresses. We extend discontinuous finite element methods for a sharp interface treatment of discontinuities in concentrations, fluxes, electric fields and in displacements, the latter arising from active particle–solid electrolyte interface fracture. We model the degradation in the charge transfer process that results from the loss of contact due to fracture at the electrolyte–active particle interfaces. Additionally, we account for the stress-dependent kinetics that can influence the charge transfer reactions and solid state diffusion. The discontinuous finite element approach does not require a conformal mesh. This offers the flexibility to construct arbitrary particle shapes and geometries that are based on design, or are obtained from microscopy images. The finite element mesh, however, can remain Cartesian, and independent of the particle goemetries. We demonstrate this computational framework on micro-structures that are representative of solid-state batteries with single and multiple anode and cathode particles.

Multi-objective optimization of sandwich structures for reinforcing composite fuselages

This study introduces a simulation-based optimization approach that combines a multi-objective genetic algorithm and the finite element method to design sandwich structures to reinforce composite fuselages. The sandwich structure has been parametrized with a set of mixed integer-continuous variables representing the polymer core thickness, the number and weight of laminated fiberglass layers, and size and position of the structure. Automatic scripting builds the geometry of the structure, then mounts it in the fuselage, and finally creates a conformal mesh. During the optimization, each reinforced fuselage is subjected to several load cases specified by the CS-22 EASA design standard, determining the worst load condition. Two objectives were considered: increasing the reinforced fuselage’s minimum fiber stress safety factor for improved safety and reducing the number of layers of the sandwich structure to minimize manufacturing costs. Structural constraints were the mass of the sandwich structure and buckling and flexo-torsional deformation of the reinforced fuselage. Results show an efficient material arrangement due to the reduced number of layers in the sandwich structure and a stress safety factor surpassing that set by the standard. Finally, the reinforcement of a glider fuselage to incorporate a retractable electric propulsion system is presented as a real-world industrial application.

A quasi-conforming embedded reproducing kernel particle method for heterogeneous materials

We present a quasi-conforming embedded reproducing kernel particle method (QCE-RKPM) for modeling heterogeneous materials that makes use of techniques not available to mesh-based methods such as the finite element method (FEM) and avoids many of the drawbacks in current embedded and immersed formulations which are based on meshed methods. The different material domains are discretized independently thus avoiding time-consuming, conformal meshing. In this approach, the superposition of foreground (inclusion) and background (matrix) domain integration smoothing cells are corrected by a quasi-conforming quadtree subdivision on the background integration smoothing cells. Due to the non-conforming nature of the background integration smoothing cells near the material interfaces, a variationally consistent (VC) correction for domain integration is introduced to restore integration constraints and thus optimal convergence rates at a minor computational cost. Additional interface integration smoothing cells with area (volume) correction, while non-conforming, can be easily introduced to further enhance the accuracy and stability of the Galerkin solution using VC integration on non-conforming cells. To properly approximate the weak discontinuity across the material interface by a penalty-free Nitsche’s method with enhanced coercivity, the interface nodes on the surface of the foreground discretization are also shared with the background discretization. As such, there are no tunable parameters, such as those involved in the penalty type method, to enforce interface compatibility in this approach. The advantage of this meshfree formulation is that it avoids many of the instabilities in mesh-based immersed and embedded methods. The effectiveness of QCE-RKPM is illustrated with several examples.

Generation of adaptive hexahedral meshes from surface and voxel geometric models

We present a modification of the developed hexahedral mesh generator from voxel data which allows constructing adaptive computational meshes. Construction of the refinement field is based on geometry features of the described model when it has a large thickness difference in dimensions or small and thin areas. A universal criterion for cells refinement is proposed which gives the possibility of its use in the case of volumetric (voxel) and surface (STL) representations of the model geometry. The refinement templates that provide conformal mesh closure are described. The results of the algorithm performance are given.

Comparative Study of Skin–Stringer Connection Approaches for Stiffened Structures with Curvilinear Stiffeners

Additive manufacturing enables the deposition of material in the near-net shape at any desired location on the base plate or shell to improve the structural performance by tailoring the buckling and/or vibration mode shapes. In this study, a stiffened plate with arbitrarily shaped stiffeners is modeled using a nonconformal mesh-based finite element model. Six degrees of freedom are considered for each node of the plate and the stiffener finite element models to enable the couplings between the plate’s and stiffeners’ in-plane and out-of-plane motions. Displacement compatibility is enforced at the interface between the plate and the stiffeners where several skin–stringer connection approaches, including inverse isoparametric mapping algorithm (IIMA), radial basis function, and thin-plate spline function, are used. Various complex models available in the literature are employed for evaluating the skin–stringer connection approaches. Research studies show that the IIMA-based nonconformal mesh modeling can generate a sparse displacement approximation matrix, which makes it an efficient approach for nonconformal mesh-based modeling. For the stiffened plate with higher stiffeners, the present results agree well with those obtained from the Abaqus/CAE conventional conformal mesh and mesh tie constraint-based nonconformal mesh modeling results, but yield better results than the Abaqus/CAE wrapping mesh approach.

A face‐centred finite volume method for high‐contrast Stokes interface problems

AbstractA new face‐centred finite volume method (FCFV) for Stokes problems involving sharp interfaces is proposed. Two formulations, based on two strong forms of the Stokes problem, and using different mixed variables, are presented. Particular attention is paid to the symmetry of the resulting system of global equations, and a simple rewriting of the interface boundary condition is proposed to ensure that one of the formulations preserves the symmetry of the linear system that is usually lost when considering material interfaces. Four numerical examples are considered to test the implementation numerically by performing mesh convergence studies, in two and three dimensions. The examples account for discontinuous viscosity as well as the effect of surface tension. The results show that one of the formulations is less sensitive to the numerical stabilisation used in FCFV methods but does not preserve the symmetry of the global system, whereas the other formulation is more sensitive to the stabilisation, but preserves the symmetry of the resulting system of equations. The FCFV method appears as a promising alternative for the simulation of viscous flow involving internal boundaries on conformal meshes. The potential application of the FCFV method for the purpose of geodynamic modelling is discussed.

Unified periodic boundary condition for homogenizing the thermo-mechanical properties of composites

This work emphasizes on the Periodic Boundary Condition (PBC) utilized in the Finite Element Homogenization (FEH) method to numerically determine the Thermo-mechanical (T-M) properties of composites. The unified numerical implementation algorithms of the PBC used for accurately predicting the elastic properties, coefficients of thermal expansion and elasto-plastic behaviors of composites and available for representative volume elements with conformal and non-conformal surface meshes, respectively, are proposed and detailed. Regarding not only the thermo-elastic properties of composites, but also the elasto-plastic behaviors of composites under both simple uniaxial tensile, shear, uniaxial cycle loading paths and complex non-radial loading paths, the unified numerical implementation algorithms are verified to accurately predict the T-M properties of composites, through comparison with the results of the analytical models, the DIGIMAT-FE method, the experimental test and the results from the literature. The unified numerical implementation algorithms of the PBC can be extended to the predictions of other properties of various composites. This work provides a comprehensive insight into the PBC used for predicting the T-M properties of composites and benefits development of future FE codes implemented in commercial software packages.

Diffuse-Interface Blended Method for Imposing Physical Boundaries in Two-Fluid Flows.

Multiphase flows are commonly found in chemical engineering processes such as distillation columns, bubble columns, fluidized beds and heat exchangers. The physical boundaries of domains in numerical simulations of multiphase flows are generally defined by a conformal unstructured mesh which, depending on the complexity of the physical system, results in time-consuming mesh generation which frequently requires user-intervention. Furthermore, the resulting conformal unstructured mesh could potentially contain a large number of skewed elements, which is undesirable for numerical stability and accuracy. The diffuse-interface approach allows for the use of a simple structured meshes to be used while still capturing the desired physical (e.g., solid-fluid) boundaries. In this work, a novel diffuse-interface method for the imposition of physical boundaries is developed for the incompressible two-fluid multiphase flow model. This model is appropriate for dispersed multiphase flows which are pervasive in chemical engineering processes, in that this flow regime results in high levels of mass and energy transfer between phases. A diffuse interface is used to define the physical boundaries and boundary conditions are imposed by blending the conservation equations from the two-fluid model with that of the nondeformable solid. The results from the diffuse-interface method are compared with results from a conformal unstructured mesh for different interface functions and widths. For small interface widths, the accuracy of the flow profile is unaffected by the choice of interface function and the phase fraction distribution and flow behavior are within 3% compared to those from a conformal mesh. As the interface width increases, the diffuse-interface solution deviates from the conformal mesh solution in both the localized gas fraction and the overall gas hold-up, resulting in a difference up to 30%. In the case of flow past a cylinder, where the solid interacts with the flow, the presence of the diffuse interface extends the thickness of the solid boundary and results in a deviation from the conformal mesh solution as time increases.

Automated conformal mesh generation chain for woven composites based on CT-scan images with low contrasts

Automated conformal mesh generation chain for woven composites based on CT-scan images with low contrasts

Direct embedment of RVE-based microscale into lab size coupons to research fracture process in unidirectional and bidirectional composites

The structural applicability of fibre-reinforced polymer (FRP) composites requires testing for their resilience against fracture. The paper presents a virtual tool to configure the microscale geometrical model constituting the representative volume element embedded in the homogeneous macroscale region, providing an alternative to the tedious experimental procedures. The conformal meshing strategies in the microregion are also elucidated in the paper. The microscale model is enriched with the constitutive material models exhibiting complex phenomena of individual constituents of the composite and cohesive damage properties. The elastoplastic and elastic–viscoplastic material models coupled with the energy-based and rate-dependent damage initiation criteria, respectively, are applied to the matrix. At the same time, the fibre-matrix interface is treated with the weak traction–separation law. Through the virtual tool, the microscale region allows to make an insightful study of the toughness mechanism and visualise the damage initiation and propagation in the FRP structure at different fracture modes.

Conformal Marked Bisection for Local Refinement of n-Dimensional Unstructured Simplicial Meshes

We present an n -dimensional marked bisection method for unstructured conformal meshes. We devise the method for local refinement in adaptive n -dimensional applications. To this end, we propose a mesh marking pre-process and three marked bisection stages. The pre-process marks the initial mesh conformingly. Then, in the first n − 1 bisections, the method accumulates in reverse order a list of new vertices. In the second stage, the n -th bisection, the method uses the reversed list to cast the bisected simplices as reflected simplices, a simplex type suitable for newest vertex bisection. In the final stage, beyond the n -th bisection, the method switches to newest vertex bisection. To allow this switch, after the second stage, we check that under uniform bisection the mesh simplices are conformal and reflected. These conditions are sufficient to use newest vertex bisection, a bisection scheme guaranteeing key advantages for local refinement. Finally, the results show that the proposed bisection is well-suited for local refinement of unstructured conformal meshes. • A new multi-stage bisection for n-dimensional unstructured conformal meshes. • Suitable for local refinement. • Leads to conformal and reflected meshes after n uniform refinements. • In the third stage, the method favors conformity, finiteness, stability, and locality.

Electric field-driven microscale 3D printing of flexible thin-walled tubular mesh structures of molten polymers

Thin-walled tubular mesh structures are the basic form of tubular scaffolds, such as vascular and nerve conduit stents, in tissue engineering. A novel electric field-driven microscale three-dimensional printing (EFD μ-3D printing) was proposed for manufacturing these structures of molten polymers with high resolution. For printing on curved substrates, the distributions of electric field force on substrates with different curvature radii in the self-excited electrostatic field were revealed via numerical simulations. The optimal process parameters for EFD μ-3D printing on curved substrates were determined. To improve printing accuracy, a micro-area preset eccentricity strategy was proposed by reducing the vertical angle of printing jets. A number of printing cases have been carried out. It is shown that the proposed method is effective for the micro-nano scale printing of 3D structures. The printed structures have good flexibility; they can be restored to their original state after 8.9 % axial compression, with an original length of 67 mm. Moreover, a conformal printing variable stiffness thin-wall tubular mesh structure has been achieved with a length of 28 mm, a line diameter of 80 μm, a big end diameter of 8 mm, and a small end diameter of 4 mm.