Tetrahedral Elements Research Articles

An enhanced polar-based GPGPU-parallelized contact detection algorithm for 3D FDEM and its application to cracking analysis of shield tunnel segmental linings

This study introduces the three-dimensional combined finite-discrete element method (3D FDEM) to perform cracking analysis of segmental linings under extreme conditions. Considering the complex contact interactions of segments, this study first proposes a polar-based GPGPU-parallelized contact detection algorithm to handle memory issues confronted by existing FDEM algorithms. Initially, a spatial decomposition approach based on the polar coordinate system is implemented during the broad search phase. Each tetrahedral element is positioned within suitable search cells according to the axis-aligned bounding box. Subsequently, element pairs within each search cell are iteratively traversed, and potential contact pairs are identified using the judge cell criteria. After the broad search, a narrow search phase is executed to determine all real contacts. Following the above implementation, a load-structure model encompassing soil spring and external pressure calculations is then proposed. Based on the shape functions and coordinate transformation, the formulations of soil springs and external loads are derived and parallelized in the 3D FDEM framework. Three numerical tests are presented to validate the effectiveness of the proposed approach. Simulation results confirm the suitability of the proposed method for cracking analysis of segmental linings. Compared to the existing methods, it reduces GPU memory usage by 56 ∼ 76 % without extending time, which enhances the computational scale of 3D FDEM simulations. Furthermore, the proposed method is applied to two engineering scenarios, i.e., straight and curved segmental linings, both considering the absence and presence of contact defects between segments. The results show that contact defects can significantly reduce the resistance of the structural system comprising bolts and concrete segments for straight and curved scenarios.

ОПТИМІЗАЦІЯ КОНСТРУКЦІЇ ТА РЕЖИМУ РОБОТИ СУШИЛЬНОЇ КАМЕРИ НА ОСНОВІ ЇЇ 3D МОДЕЛІ

This paper focuses on the analysis and optimization of the drying chamber design using SolidWorks Simulation software. One of the main problems that arises in the design of drying chambers is the need to ensure optimal conditions for efficient drying of materials. In this regard, this study determines the optimal design parameters of the drying chamber and its operating mode to ensure uniform temperature distribution in the stack. The methodology includes creating a 3D model of the drying chamber and its main components in SolidWorks, generating a difference mesh, conducting a thermal study in SolidWorks Simulation, and applying Design Study to optimize the design parameters of the drying chamber and its operating modes. The simulation results include the output of the heat flow in the stack and temperature measurements at various points, which allows us to assess the temperature differences and the impact of optimization parameters on them. Temperature measurements at different points were performed using the Prob tool, which takes into account the exact geometry and location of nodes on the tetrahedral elements of the difference grid of the 3D model of the drying chamber used for the calculations. In general, the study consists of applying an integrated approach to the design of drying chambers using SolidWorks and automating the optimization process through the SolidWorks API, for which software has been created. The software also allows users to enter input data for the 3D model design and thermal study parameters for their transfer to SolidWorks. In addition, the software provides the ability to perform calculations directly in SolidWorks Simulation but in the background, so that the results obtained will be displayed in a special program window in a fully automated mode. It is also worth noting that performing calculations without visualizing this process significantly reduces the time to obtain results that can be used to improve the technological processes of drying various materials in drying chambers.

Modeling the failure process of rock masses using a 3D nodal-based continuous-discontinuous deformation analysis method

A three-dimensional nodal-based continuous-discontinuous deformation analysis method (3D-NCDDAM) is developed in this study for modeling the failure process of rock masses. In the 3D-NCDDAM, four-node tetrahedral elements which can be automatally generated are used to discretize the problem domain. To reduce computational cost, and effectively model the failure process of rock masses at concerned regions, the problem domain is devided into two types of regions, namely, the “non-destructive region” and “possible destructive region”. The non-destructive region is mainly used to reduce the boundary effects on the numerical results, and is regarded transitional zone in which the failure of rock masses is not considered. For the possible destructive region, artificial joint elements are inserted, and the failure of these joint elements is used to represent fracture initiation and propagation in rock masses. Additionally, the contact potential which is initially proposed in finite-discrete element method (FDEM) is incorporated into the 3D-NCDDAM to simuate the opening and slipping of discontinuous structural planes involved in rock masses’ failure process. Note that the difficultly in determing contact types in the traditional 3D discontinuous deformation analysis (3D-DDA) method is avoided in the 3D-NCDDAM. The proposed 3D-NCDDAM can be treated as a hybrid method, which inherite both the advantage of DDA method and FDEM. To validate the 3D-NCDDAM, seven numerical examples are conducted. The numerical results indicte that the 3D-NCDDAM can effectively model the failure process of rock masses in both experimental scale and engineering scale.

Stability of Proximal Femoral Osteotomies in Pediatric Bone Models Fixed with Flexible Intramedullary Nails and Evaluated by the Finite Element Method.

Objective To evaluate the stability of osteotomies created in the subtrochanteric and trochanteric regions in a pediatric femur model fixed by flexible intramedullary rods. Method Tomographic sections were obtained from a pediatric femur model with two elastic titanium rods and converted to a three-dimensional model. This model created a mesh with tetrahedral elements according to the finite element method. Three virtual models were obtained, and osteotomies were performed in different regions: mediodiaphyseal, subtrochanteric, and trochanteric. A vertical load of 85N was applied to the top of the femoral head, obtaining the displacements, the maximum and minimum main stress, and the equivalent Von Mises stress on the implant. Results With the applied load, displacements were observed at the osteotomy site of 0.04 mm in the diaphyseal group, 0.5 mm in the subtrochanteric group, and 0.06 mm in the trochanteric group. The maximum stress in the diaphyseal, subtrochanteric, and trochanteric groups was 10.4 Pa, 7.52 Pa, and 26.4 Pa, respectively. That is around 40% higher in the trochanteric group in regards to the diaphyseal (control). The minimum stress of the bone was located in the inner cortical of the femur. The equivalent Von Mises stress on the implants occurred at osteotomy, with a maximum value of 27.6 Pa in the trochanteric group. Conclusion In both trochanteric and subtrochanteric osteotomies, fixation stability was often lower than in the diaphyseal model, suggesting that flexible intramedullary nails are not suitable implants for proximal femoral fixations.

Low-order finite element method for the static and vibration analysis of an antenna pedestal

In this work, an effective algorithm is developed for the static and vibration analysis of antenna pedestal, which can well balance the modelling efficiency and computational accuracy. At first, the antenna pedestal is discretized into a set of low-order tetrahedral elements that can be automatically generated. Then, the face-based smoothing domains are constructed based on the faces of the low-order tetrahedral mesh. With the help of strain smoothing technique, the system stiffness matrices are calculated in these smoothing domains and finally, the static and vibration responses of the antenna pedestal are obtained based on the related system equations. There is no need to add any penalty parameters or additional degrees of freedom. Numerical results demonstrate that the present algorithm can achieve much better accuracy and higher convergence rate than traditional FEM, which can be regarded as an efficient tool for the structural analysis of antenna pedestal.

Finite element analysis (FEA) of stress distribution in platform-switched short dental implants.

The distribution of stress on short platform switched dental implants is of interest. Hence, the mandibular posterior molar area was modelled using a three-dimensional finite element method (FEM) with a continuous 1.5 mm cortical bone thickness and an inner cancellous bone core. The implants used in the study were 5 mm long, 4.5 mm wide and 3.5 mm wide at the abutments. 120 N of force was applied in both the vertical and oblique (20° and 35°) directions to create a realistic simulation. ANSYS Workbench was generated for each model. Von Mises stress was assessed in the cortical and cancellous bones at varying depths. Ten noded tetrahedron elements with three degrees of freedom per node were employed to interpret translations on the x, y, and z axes. The stress-based biomechanical behaviour of platform switched short osseo-integrated implants varied across all 5 positions in FEM simulations, based on the depth of implant placement, the direction of applied force, and the shape of the bone. Data shows that opposite forces to the vertical forces caused more damage. Thus, the implantation of subcrestal implants resulted in reduced stress on the cortical and cancellous bone.

Body‐fitted topology optimization via integer linear programming using surface capturing techniques

AbstractRecent advancements in computational tools and additive manufacturing have expanded design possibilities on fluid devices and structures to also include aesthetics. However, traditional discrete density‐based topology optimization methods usually use square and cubic regular meshes, resulting in jagged contour patterns that require mesh refinement and post‐processing to numerical solution steps of complex physics problems. In this article, we propose a new isosurface boundary capture strategy for topology optimization in structural and fluid flow problems. The capture of smoothed boundaries is done via a simple strategy through element splitting and analysis domain remeshing. The smoothing procedure novelty lies on the optimizer regular mesh decomposition into smaller triangular or tetrahedral elements with a pseudo‐density nodal function that produces implicit geometry boundaries. We employ the TOBS‐GT (topology optimization of binary structures with geometry trimming) method to solve optimization problems for mean compliance and fluid flow energy dissipation, subject to a volume fraction constraint. Since, we perform discrete body‐fitted optimization, the material interpolation models are expressed in linear form without penalty factors. Our method produces a lower computational cost topology evolution with high resolution and mitigated sharp details, achieved via smooth edges and surfaces, as demonstrated through benchmark numerical examples in two‐ and three‐dimensional space. In addition, the proposed strategy facilitates the optimization of benchmark fluid flow examples for moderate Reynolds numbers.

Optimization of a tetrahedron compliant spherical joint via computer-aided engineering tools

This article focuses on enhancing the range of motion (ROM) of the Tetra II joint, a spherical compliant joint consisting of three internally interconnected tetrahedron-shaped elements that achieve motion through elastic deformation. Despite its excellent precision, this specific design is constrained in terms of ROM due to internal contacts among the tetrahedral elements. To overcome this limitation, this study utilizes a computer-aided engineering (CAE) framework to optimize the configuration of the Tetra II joint and enhance its ROM. The resultant optimized joint, referred to as Tetra III, is subsequently compared to Tetra II in terms of both ROM and center shift. Finite element models (FEM) are employed to validate the optimization results and examine how various tetrahedron-shaped geometries impact the joint’s performance. The newly optimized joint exhibits a significantly higher ROM compared to the previous version, while maintaining excellent precision and overall smaller dimensions. Finally, to demonstrate its manufacturability, the Tetra III joint is produced using selective laser sintering (SLS) technology, with Duraform PA serving as the construction material. The successful fabrication serves as a demonstrative example of the improved design of the Tetra III joint.

Raster approach to modelling the failure of arbitrarily inclined interfaces with structured meshes

This paper presents an approach to evaluate the failure of arbitrarily inclined interfaces using FE models with structured spatial discretization, providing accurate prediction of crack propagation along paths known a priori that are not constrained to the element boundaries. The combination of algorithms for the generation of structured discretization of representative polycrystalline microstructures with novel cohesive element formulations allow modelling the failure of complex topologies along rasterised boundaries, with noticeably higher computational efficiency and comparable accuracy. Two formulations of raster cohesive elements are presented, adopting either elastic-brittle or Tvergaard–Hutchinson traction separation laws. The formulations proposed are first validated comparing the failure of the interface within bi-crystal structures discretised using hexahedral elements either within a structured mesh (i.e. with rasterised boundaries) or an unstructured mesh (i.e. with planar boundary). Subsequently, the effectiveness of the formulations is demonstrated comparing the inter-granular crack propagation within complex polycrystalline microstructures. The behaviour of the novel cohesive element formulation in structured meshes consisting of regular hexahedral elements is in excellent agreement with the deformation and failure of classic cohesive element formulations placed along the planar boundaries of unstructured meshes consisting of tetrahedral elements. The higher computational cost of the raster cohesive elements is more than compensated by the increase in computational efficiency of structured meshes when compared to unstructured meshes, leading to a reduction of the simulation time of up to over 200 times for the simulations presented in the paper, thus allowing the simulation of large domains.

GPU parallel computation strategy for electrothermal coupling problems using improved assembly-free FEM

Abstract The analysis of electrothermal coupling problems finds extensive application in engineering. However, for large-scale electrothermal coupling problems, the time cost and storage requirements for solving them using the finite element method (FEM) are substantial. We optimize the finite element electrothermal coupling computation from two aspects: computational speed and storage usage. Based on the assembly-free FEM, we explore the symmetry of element matrices to reduce storage for second-order tetrahedral elements and propose a graphics processing unit (GPU) parallel algorithm to improve computational speed. At the same time, we allocate the parallel parts of an electrothermal coupling problem to two GPUs to improve the speed further. In addition, for the three types of boundary conditions in electrothermal coupling problems, we design parallel application methods suitable for assembly-free FEM. Finally, we compare our strategy with methods from other literature through the numerical experiment. Our method reduces the element matrices’ storage by 45%. Compared with the solution process using the element level method and degree of freedom level method, our strategy achieves average acceleration ratios of 5.83 and 1.38, respectively.

Fluid-evolving landform interaction by a surface-tracking method

This paper introduces a continuous finite element model to simulate fluid flow-bedform interaction problems. The approach utilizes a non-oscillatory finite element algorithm to compute the fluid dynamics by solving the complete Navier–Stokes equations. Additionally, it addresses the evolution of the fluid–bedform interface as a consequence of spatially non-balanced sediment fluxes through the solution of a conservation equation for the erodible layer thickness. A sign preservation algorithm is particularly relevant for landform tracking because a positive definite thickness of the erodible sediment layer is essential to model the interaction between evolving cohesionless sediment layers and rigid beds. The fluid/terrain interface is explicitly captured through a surface tracking methodology. First, new nodes fitting the interface are incorporated into the finite element mesh; then, elements beneath this interface are deactivated, while intersected elements are restructured to get a mesh composed exclusively of tetrahedral elements. Numerical experiments demonstrate capabilities of the method by exploring relevant problems related with civil engineering, such as the evolution of trenches and the scour of a submerged pile.

Three-dimensional thermoelastic analysis of functionally graded solids using a high-order staggered finite volume method

This paper has developed a 3D 2-order cell vertex finite volume method (CV-FVM) for thermoelastic problems of functionally graded materials (FGMs). The construction methods of 15-node triangular prismatic (P15) element and 20-node hexahedral (H20) element are proposed. The 10-node tetrahedral (T10) element is adopted together with P15 and H20 to deal with mixed element problems. Material variation is considered using the staggered grid technique by defining properties at the cell center while defining unknown variables at the node. This treatment makes the present method suitable for FGMs with arbitrary material distribution. The accuracy and capability of the 2-order CV-FVM are verified by numerical tests. The results show that all the T10 elements, P15 elements, H20 elements and 2-order mixed elements can provide accurate results. When the same mesh size is employed, 2-order CV-FVM could provide better numerical precision than the 1-order CV-FVM and H20 elements could provide more accurate results than T10 elements. Based on the same mesh size, the H20 element avoids numerical oscillations caused by the T10 element. The developed method exhibits advantages in analyzing materials with significant property changes, avoiding physically unrealistic stress distributions.

Numerical Integration of Some Arbitrary Functions over an Ellipsoid by Discretizing into Hexahedral Elements for Biomaterial Studies

This study mathematically examines chemical and biomaterial models by employing the finite element method. Unshaped biomaterials’ complex structures have been numerically analyzed using Gaussian quadrature rules. It has been analyzed for commercial benefits of chemical engineering and biomaterials as well as biorefinery fields. For the computational work, the ellipsoid has been taken as a model, and it has been transformed by subdividing it into six tetrahedral elements with one curved face. Each curved tetrahedral element is considered a quadratic and cubic tetrahedral element and transformed into standard tetrahedral elements with straight faces. Each standard tetrahedral element is further decomposed into four hexahedral elements. Numerical tests are presented that verify the derived transformations and the quadrature rules. Convergence studies are performed for the integration of rational, weakly singular, and trigonometric test functions over an ellipsoid by using Gaussian quadrature rules and compared with the generalized Gaussian quadrature rules. The new transformations are derived to compute numerical integration over curved tetrahedral elements for all tests, and it has been observed that the integral outcomes converge to accurate values with lower computation duration.

Influence of the Curve of Spee on Tooth Displacement Patterns: A Finite Element Analysis at Varying Implant Heights.

Background Monocortical mini-screw-type temporary anchorage devices (TADs), or mini-screws, have significantly impacted orthodontic treatment strategies, especially in severe crowding and protrusion cases. These devices offer flexibility in placement sites, but the chosen location can considerably influence tooth displacement patterns. Key factors include the 'line of force' and the biomechanical properties of orthodontic tools. By analyzing tension distribution and three-dimensional displacements, the finite element method (FEM) provides a thorough means to comprehend these patterns. The Curve of Spee (COS) is a crucial factor potentially affecting displacement. Objective This study aimed to leverage finite element analysis (FEA) to understand the impact of varying mini-implant heights (10 mm, 13 mm, and 16 mm) on the displacements of different tooth types under a consistent force of 150 gm and compare these displacements both in the presence and absence of the COS. Materials and methods A CAD model of the jaw and teeth was developed using CT scan data and a Rexcan III 3D White Light Scanner. This model was meshed in Altair HyperMesh using tetrahedral elements, resulting in a Finite Element Model. The model incorporated various components, including teeth, the periodontal ligament (PDL), alveolar bone, brackets, a titanium mini-screw, and an archwire measuring 0.019 x 0.025 inches. Unique material properties were assigned to the PDL, and the assembly accurately replicated the clinical alignment of the archwire and brackets. Subsequently, stress and strain analyses were conducted on the model using the FEM. Results The displacement patterns of various teeth at implant heights of 10 mm, 13 mm, and 16 mm under a 150-gm force were analyzed in relation to the COS. Notably, for the central incisor, the COS significantly affected displacements in the Y and Z directions. Similarly, the Lateral Incisor and Canine exhibited marked changes in the Z direction with the presence of the COS. The Second Premolar's apex displacement showed significant variation due to the COS, while the First Molar displayed notable changes in the X direction. Generally, the presence of the COS either maintained or slightly increased Z-directional displacements across teeth, particularly at the apices. Conclusion The presence of COS significantly influences tooth displacement patterns when using mini-screws at different implant heights. Central incisors, lateral incisors, and canines are particularly sensitive to changes in the Z direction with the COS. The biomechanical analysis emphasizes the importance of considering COS in treatment planning for optimal results with mini-implants in orthodontics.

FEA Comparison of the Mechanical Behavior of Three Dental Crown Materials: Enamel, Ceramic, and Zirconia.

The restoration of endodontically treated teeth is one of the main challenges of restorative dentistry. The structure of the tooth is a complex assembly in which the materials that make it up, enamel and dentin, have very different mechanical behaviors. Therefore, finding alternative replacement materials for dental crowns in the area of restorative care isa highly significant challenge, since materials such as ceramic and zirconia have very different stress load resistance values. The aim of this study is to assess which material, either ceramic or zirconia, optimizes the behavior of a restored tooth under various typical clinical conditions and the masticatory load. A finite element analysis (FEA) framework is developed for this purpose. The 3D model of the restored tooth is input into the FEA software (Ansys Workbench R23)and meshed into tetrahedral elements. The presence of masticatory forces is considered: in particular, vertical, 45° inclined, and horizontal resultant forces of 280 N are applied on five contact points of the occlusal surface. The numerical results show that the maximum stress developed in the restored tooth including a ceramic crown and subject to axial load is about 39.381 MPa, which is rather close to the 62.32 MPa stress computed for the natural tooth; stresses of about 18 MPa are localized at the roots of both crown materials. In the case of the zirconia crown, the stresses are much higher than those in the ceramic crown, except for the 45° load direction, while, for the horizontal loads, the stress peak in the zirconia crown is almost three times as large as its counterpart in the ceramic crown (i.e., 163.24 MPa vs. 56.114 MPa, respectively). Therefore, the zirconia crown exhibits higher stresses than enamel and ceramic that could increase in the case of parafunctions, such as bruxism. The clinician's choice between the two materials should be evaluated based on the patient's medical condition.

An accurate coupled method for analysis of transcranial magneto-acoustic-electrical stimulation

In this paper, a numerical method for simulating transcranial magneto-acoustic-electrical stimulation (TMAES) is presented. The smoothed finite element method (SFEM) is applied to calculate the magnetic field and the Rayleigh integral (RI) is used for solving the ultrasonic field. Three-node triangular and four-node tetrahedral elements are set as the background cells for constructing the smoothing domains (SDs). A gradient smoothing technique (GST) is applied over each SD to obtain the magnetic flux density. The discretized system equations are obtained by using the generalized smoothed Galerkin weakform. To solve the sound pressure, the ultrasonic transducer is discretized into a series of micro-elements. The acoustic pressure in the problem domain is then equal to the superposition of that generated by the individual micro-element. Finally, the current density can be calculated by the magneto-acoustic coupling effect. Numerical examples demonstrate that SFEM can improve accuracy, convergence rate, and efficiency to 10, 1.8, and 4 times those of FEM, respectively. In the calculation of coupled fields, SFEM-RI also improves the accuracy by 2.5 times. Additionally, the presented method possesses the property of insensitivity to mesh distortion.

Design and Analysis of UAV Profile for Agriculture and Surveying Application

This study represents the aerodynamic design of an Unmanned aerial vehicle intended for surveillance or agriculture with a maximum take weight of 125 kg. Weight estimation and constraint analysis were done based on the Mission profile. Design of Computer-Aided Design (CAD) models were generated for three different configurations using CATIA V5R20 as a high wing, mid-wing, and low wing. Flow analysis was done for the above configurations at various angles of attack. ANSYS 15 was used for the flow Analysis. A Tetrahedron element meshed the model with the minimum required orthogonal quality. Five microns were given to the initial layer height of the prism mesh. Spalart Allmaras model is used as the Turbulence model in the solver. The aerodynamic characteristics of the above configuration obtained from Computational Fluid Dynamics (CFD) results were compared with the DATCOM program and validated with the wind tunnel experimental test data. The open-circuited suction-type Subsonic wind tunnel was employed for the test. The aerodynamic properties for the angle of attack in the range of -2° to 14° angle of attack are calculated using a six-component balance. The study aims to find the Unmanned Aerial Vehicle (UAV) configuration based on the aerodynamic characteristics obtained from the CFD and DATCOM results. High-wing UAVs have better aerodynamic efficiency than the other two configurations.

Thermal Analysis of Concrete Blocks and Stack-Bond Prisms under Different Boundary Conditions

Fire is a significant threat to human lives and the integrity of buildings. To better understand the complex behavior of masonry exposed to high temperatures, thermal analyses were carried out to evaluate the temperature distribution in concrete blocks and stack-bond prisms exposed to high temperature levels. The effects of distinct specimen boundary conditions (restrict or easy access to air circulation inside the voids of the block and prisms) on the thermal response of the masonry materials were investigated. Thersys 2.0 software was used to implement three-dimensional thermal analysis of distinct finite element models. Four-node tetrahedral elements and full integration were used in all models. The modeling approach was validated by experimental data obtained from thermocouples embedded into masonry components. The results indicated that the boundary conditions significantly affected the time required for homogenization of temperature in blocks and prisms. Easy access to air circulation inside the voids of the prisms provided a faster temperature homogenization. In this scenario, the prism reached temperature ranges of (300 ± 0.5% × 300) °C and (600 ± 0.5% × 600) °C after exposure times of 2 h and 2 h 10 min, respectively. When access to air circulation within the voids of the prisms was limited, the same temperature ranges were achieved after exposure times of 5 h 20 min and 6 h, respectively.

Computer-Assisted Modeling of Greater Tubercle Transposition Effect on Stress Distribution in Shoulder

Background. The aim of the work was to investigate the influence of the position of a fractured greater tubercle of the humerus on stress distribution in the glenohumeral joint (GHJ) at shoulder abduction by performing stress-strain analysis (SSA). Material and methods. A simulated 3D model of the shoulder was constructed on the basis of anatomical and anthropometric data as close as possible to real ones using the SolidWorks software package.Results. A finite-element model (FEM) was generated in a semi-automatic mode composed mostly of tetrahedral and hexagonal elements < 1 mm in dimensions.Conclusion. The simulated 3D model of the shoulder, based on anatomical and anthropometric data as close as possible to the real ones, makes possible highly accurate determination of the optimal mutual position of the movable elements of the real joint individually for each specific patient, and makes it possible to accurately perform a minimally invasive surgical operation.

Evaluation of bone mineral density in adolescent idiopathic scoliosis using a three-dimensional finite element model: a retrospective study

BackgroundAdolescent idiopathic scoliosis (AIS) is often accompanied by osteopenia and osteoporosis, which can cause serious complications. The aim of this study was to determine the specific bone mineral density (BMD) of each vertebral body in patients with AIS using biomechanical finite element modeling based on three-dimensional (3D) reconstruction.MethodsThis retrospective study involved 56 patients with AIS. Computed tomography (CT) and radiography were performed. Spinal vertebrae were segmented from the spinal CT images of patients with AIS to reconstruct 3D vertebral models. The vertebral models were meshed into tetrahedral finite elements to assess the BMD.ResultsThe mean main curve Cobb angle was 88.6 ± 36.7°, and the mean kyphosis angle was 36.8 ± 31.5°. The mean BMD of the global spine was 0.83 ± 0.15 g/cm2. The highest BMD was measured on the concave side of the apex (0.98 ± 0.16 g/cm2). Apical vertebral BMD was negatively correlated with age and height (r = − 0.490, p = 0.009 and r = − 0.478, p = 0.043, respectively). There were no significant differences in BMD values between the concave and convex sides (p > 0.05).ConclusionsThe 3D finite element modeling of BMD in patients with AIS is a reliable and accurate BMD measurement method. Using this method, the overall BMD of patients with AIS was shown to gradually decrease from the top to the bottom of the spine. Our findings provide valuable insights for surgical planning, choice of screw trajectories, and additional biomechanical analyzes using finite element models in the context of scoliosis.