Free-form Deformation Technique Research Articles

Computational Design Optimization for S-Ducts

In this work, we investigate the computational design of a typical S-Duct that is found in the literature. We model the design problem as a shape optimization study. The design parameters describe the 3D geometrical changes to the shape of the S-Duct and we assess the improvements to the aerodynamic behavior by considering two objective functions: the pressure losses and the swirl. The geometry management is controlled with the Free-Form Deformation (FFD) technique, the analysis of the flow is performed using steady-state computational fluid dynamics (CFD), and the exploration of the design space is achieved using the heuristic optimization algorithm Tabu Search (MOTS). The results reveal potential improvements by 14% with respect to the pressure losses and by 71% with respect to the swirl of the flow. These findings exceed by a large margin the optimality level that was achieved by other approaches in the literature. Further investigation of a range of optimum geometries is performed and reported with a detailed discussion.

Constrained airfoil optimization using the area-preserving free-form deformation

Purpose The purpose of this paper is the presentation of a technique to be integrated in a numerical airfoil optimization scheme, for the exact satisfaction of a strict equality cross-sectional area constraint. Design/methodology/approach An airfoil optimization framework is presented, based on Area-Preserving Free-Form Deformation (AP FFD) technique. A parallel metamodel-assisted differential evolution (DE) algorithm is used as an optimizer. In each generation of the DE algorithm, before the evaluation of the fitness function, AP FFD is applied to each candidate solution, via coupling a classic B-Spline-based FFD with an area correction step. The area correction step is achieved by solving a sub problem, which consists of computing and applying the minimum possible offset to each one of the free-to-move control points of the FFD lattice, subject to the area preservation constraint. Findings The proposed methodology is able to obtain better values of the objective function, compared to both a classic penalty function approach and a generic framework for handling constraints, which suggests the separation of constraints and objectives (separation-sub-swarm), without any loss of the convergence capabilities of the DE algorithm, while it also guarantees an exact area preservation. Due to the linearity of the area constraint in each axis, the extraction of an inexpensive closed-form solution to the sub problem is possible by using the method of Lagrange multipliers. Practical implications AP FFD can be easily incorporated into any 2D shape optimization/design process, as it is a time-saving and easy-to-implement repair algorithm, independent from the nature of the problem at hand. Originality/value The proposed methodology proved to be an efficient tool in facing airfoil design problems, enhancing the rigidity of the optimal airfoil by preserving its cross-sectional area to a predefined value.

An Automatic Aerodynamic Shape Optimisation Framework Based on DAKOTA

Aerodynamic design is a must and fundamental aspect for better performance of an aircraft. This work describes a framework for aerodynamic shape optimisation (ASO) based on high-fidelity Computational Fluid Dynamics (CFD) analyses and an open source optimisation toolkit DAKOTA. The free form deformation (FFD) technique is adopted for parameterization of geometries, followed by a transfinite interpolation (TFI) method for grid deformation. CFD simulations are performed by full Navier-Stokes equations in conjunction with turbulence model. CFD, FFD and TFI modules are integrated with DAKOTA by a black-box interface, a pre-processing and a post-processing. The transonic RAE2822 airfoil is employed for demonstration of the effectiveness and capability of the developed framework. Several cases with different optimisation algorithms, design variables and turbulence models are performed. Optimum solutions with significant improvement of lift-drag ratio are presented to demonstrate the capability of the framework in this paper.

A finite element muscle building block derived from 3D ultrasound: application to the human gastrocnemius

Muscle fascicle structure characterises muscle function, which in turn plays a key role in computer simulation of muscle shape. In this study, we use 3D ultrasound from the human medial gastrocnemius muscle to identify and map the muscle fascicle orientation and deformation during passive motion in four subjects. This muscle fascicle description is integrated into a representative muscle volume element using a free-form deformation technique to create a muscle primitive building block that deforms according to the embedded muscle fascicles within. For each subject computed passive tensile force was used to optimise the constitutive behaviour so that the known deformation matched this load. Each subject was fit to match deformation at 25, 50, 75 and 100% of muscle stretch. The subjects that exhibited a larger muscle fascicle pennation angle change during passive stretch were characterised best by a less stiff constitutive law. In contrast, subjects that had a more parallel muscle fascicle deformation, showed stiffer behaviour. The medial gastrocnemius muscle built from these muscle primitives exhibited a contractile shape that is consistent to that observed in human gastrocnemius contraction. This shape was evaluated against the same muscle embedded with muscle fascicles derived from diffusion tensor magnetic resonance imaging and was in good qualitative agreement. Muscle principal strain was shown to align with muscle fascicle direction and was spatially non-uniform. Muscle primitives may be used as building blocks to build large muscle volumes for mechanics simulation, visualisation and medical education.

Computer-assisted design and finite element simulation of braces for the treatment of adolescent idiopathic scoliosis using a coronal plane radiograph and surface topography

BackgroundOrthopedic braces made by Computer-Aided Design and Manufacturing and numerical simulation were shown to improve spinal deformities correction in adolescent idiopathic scoliosis while using less material. Simulations with BraceSim (Rodin4D, Groupe Lagarrigue, Bordeaux, France) require a sagittal radiograph, not always available. The objective was to develop an innovative modeling method based on a single coronal radiograph and surface topography, and assess the effectiveness of braces designed with this approach. MethodsWith a patient coronal radiograph and a surface topography, the developed method allowed the 3D reconstruction of the spine, rib cage and pelvis using geometric models from a database and a free form deformation technique. The resulting 3D reconstruction converted into a finite element model was used to design and simulate the correction of a brace. The developed method was tested with data from ten scoliosis cases. The simulated correction was compared to analogous simulations performed with a 3D reconstruction built using two radiographs and surface topography (validated gold standard reference). FindingsThere was an average difference of 1.4°/1.7° for the thoracic/lumbar Cobb angle, and 2.6°/5.5° for the kyphosis/lordosis between the developed reconstruction method and the reference. The average difference of the simulated correction was 2.8°/2.4° for the thoracic/lumbar Cobb angles and 3.5°/5.4° the kyphosis/lordosis. InterpretationThis study showed the feasibility to design and simulate brace corrections based on a new modeling method with a single coronal radiograph and surface topography. This innovative method could be used to improve brace designs, at a lesser radiation dose for the patient.

Aerodynamic shape optimization of an airliner elastic wing

Aerodynamic shape optimization of a common airliner elastic wing is presented and compared with optimization of the same wing assuming rigid structure. A gradient-based optimization approach is applied in connection with adjoint method used for calculation of gradients of aerodynamic forces. The initial wing shape is parameterized by Free Form Deformation technique. The aerodynamics of the elastic wing is calculated using the fluid-structure interaction method which couples computational fluid dynamic solver with linear structural solver.

Concurrent editing of automotive styling and structure with wireframe-pair

A constant demand in engineering design is to couple aesthetics and functionality of design. In this paper, a method is proposed to rapidly update the structural analysis models given the modifications made in the styling/packaging designs, forming a concurrent mechanism for developing the automotive styling and body structure. The seemingly disparate domains are represented by and thus coupled through a pair of wireframes. A joint mapping based on the hard points in packaging and a part mapping encoding local influences between wireframes are established. The deformation transfer from one of the wireframes to the other is solved as an optimization with constraints obtained from the two mappings. Finally, the finite element mesh model is adapted in accordance with the deformed structural wireframe, using a mesh morphing method based on the free-form deformation technique. Numerical results validate the proposed method and demonstrate its effectiveness.

Multiobjective shape optimization of a hypersonic lifting body using a correlation-based transition model

Although substantial numbers of aerodynamic shape optimization works have been carried out in the past few decades, the effects of boundary layer transition are not considered in the overwhelmingly majority of those previous studies. For more accurate prediction of the flow field, and for exploration of relations between aerodynamic heating and geometrical features, the commonly used local correlation-based transition model [Formula: see text] is implemented into a computational fluid dynamics (CFD) solver. A CFD-based multiobjective shape optimization method is then described and applied to a hypersonic lifting body in this paper. For parametric modeling and grid deformation, the flexible free form deformation and robust transfinite interpolation techniques are used respectively. The response surface model is adopted as the approximation model for the reduction of computational cost, and the multiobjective genetic algorithm is employed for the exploration of optimal solutions. Optimization processes have been conducted by the optimization toolkit DAKOTA and two Pareto fronts are obtained. Besides, typical solutions on the Pareto fronts are compared and analyzed. The results obtained from the case considered in this study indicate that the lift-to-drag ratio and transition onset position are closely related with the volume of the vehicle, and trade-off can be made for aerodynamic and aerothermodynamic performance from the optimal solutions. Moreover, the design problem of lifting body demonstrates the robustness and practicality of the new optimization procedures, which incorporates the transition phenomenon for hypersonic flow.

Computational design of an automotive twist beam

Abstract In recent years, the automotive industry has known a remarkable development in order to satisfy the customer requirements. In this paper, we will study one of the components of the automotive which is the twist beam. The study is focused on the multicriteria design of the automotive twist beam undergoing linear elastic deformation (Hooke's law). Indeed, for the design of this automotive part, there are some criteria to be considered as the rigidity (stiffness) and the resistance to fatigue. Those two criteria are known to be conflicting, therefore, our aim is to identify the Pareto front of this problem. To do this, we used a Normal Boundary Intersection (NBI) algorithm coupling with a radial basis function (RBF) metamodel in order to reduce the high calculation time needed for solving the multicriteria design problem. Otherwise, we used the free form deformation (FFD) technique for the generation of the 3D shapes of the automotive part studied during the optimization process. Highlights We model the automotive twist beam. We developed an algorithm to solve multicriteria optimization problem. We solve our industrial problem with our developed algorithm. New profiles of the twist beam are proposed.

Efficient measurement of aero-engine blade considering uncertainties in adaptive machining

Blade geometry plays an extremely important role in the aerodynamic performance of aero-engine. However, in the adaptive machining process of a near-net-shape blade, it is important for an application engineer to reconstruct an actual profile with high-precision measurement data in a short time. This paper presents an efficient measurement to meet the requirement for digital information in the adaptive machining, which is also intimately associated with the previous work in reconstruction (Zhang et al. in Int J Adv Manuf Technol 2015:1–16, 2015). Comparing with conventional measurement, the proposed method can not only improve the utilization of measurement data from multi-sensor system but also consider the uncertainties of measurement and localization in data merging. Concretely speaking, the low-precision data from non-contact measurement device is used for establishing possible substitute models. The modified Bayesian information (Modi_BIC) criterion based on Bayesian statistics is introduced as the principle to select the model structure. In order to reduce misalignment errors of the selected substitute model, the statistical analysis of Fisher information matrix is applied to determine registration points. Based on this, the uncertainties can be given in the form of variation boundaries. Comparisons with the specified tolerance band determine a set of re-measured points by using a contact measurement device. The high-precision data can be merged with the selected substitute model by means of free form deformation (FFD) technique. The final measurement data is provided in the form of skinning surface that is consistent to the reverse modeling methodology of aero-engine blade. Moreover, some practical examples are given to demonstrate the effectiveness and superiority of the proposed method.

基于直接控制FFD参数化方法的跨声速层流翼身组合体稳健性设计

The geometric shape is not directly manipulated by free-form deformation (FFD) technique. A high level FFD control volume is always adopted in the refined aerodynamic design. The direct manipulated free-form deformation (DFFD) technique is developed in this paper to improve the shortcomings of the FFD technique. This method adopt a parameter identification technique which can identify the disturbance quantities and convert them into displacement values on the FFD control lattices. The DFFD technique can accomplish refined aerodynamic shape design using less design variables. The visual impression is much better than the traditional FFD technology, and the number of the design variables is not increased with using of the high level FFD control volume. The robust design system is established by DFFD technique, multi-groups collaboration particle swarm optimization (MCPSO) and adaptive sampling algorithm, combined with quaternion mesh deforming technique and the numerical simulation of boundary layer transition based on γ - Re θt transition model. A transonic laminar wing body is optimized by this system. The design results show that not only a wide range of laminar flow area is maintained, but also the development of shock is controlled.

WE-D-303-05: Dosimetric Accuracy of CBCT Images Estimated by a Motion Modeling and Free-Form Deformation Technique for Radiotherapy of Lung Cancer

Purpose: To investigate the dosimetric accuracy of CBCTs estimated by a motion modeling and free-form deformation(MM-FD) technique for radiotherapy of lung cancer. Methods: Various inter-fractional variations featuring patient motion pattern change, tumor size change and tumor average position change were simulated from planning-CT to on-board images using both digital and physical motion phantoms. The doses calculated on the planning-CT (planned doses), the on-board CBCT estimated by MM-FD (MM-FD doses) and the on-board CBCT reconstructed by the conventional Feldkamp-Davis-Kress(FDK) algorithm (FDK doses) were compared to the on-board dose calculated on the ‘gold-standard’ on-board images (gold-standard doses). The absolute deviations of minimum dose (dDmin), maximum dose (dDmax), mean dose (dDmean) and dose coverage (dV100%) of PTV were evaluated. In addition, 4D on-board dose accumulations were performed using the 4D-CBCT images estimated by MM-FD. The accumulated doses were compared to measurements using OSL detectors and radiochromic films. Results: Of all the 50 scenarios simulated, the average(± standard deviation) dDmin, dDmax, dDmean and dV100% (values normalized by the prescription dose or the PTV volume) between the planned and the gold-standard PTV doses were 34.8% (± 29.2%), 3.2% (± 3.8%), 3.5% (± 3.5%) and 13.0% (± 11.4%), respectively. The corresponding values of FDK PTV doses were 3.1% (± 3.7%), 1.4% (± 1.1%), 2.1% (± 0.8%) and 14.5% (± 14.2%), respectively. In contrast, the corresponding values of MM-FD PTV doses were 0.4% (± 0.5%), 0.9% (± 0.7%), 0.6% (± 0.4%) and 0.9% (± 0.8%), respectively.For the 4D dose accumulation study, the average(± standard deviation) absolute dose deviation (normalized by local doses) between the accumulated doses and the OSL measured doses was 3.0% (± 2.4%). The average gamma pass-rate(3%/3mm) between the accumulated doses and the radiochromic film measured doses was 96.1%. Conclusion: MM-FD estimated CBCT enables accurate on-board dose calculation and accumulation for lung radiation therapy. The research was funded by the National Institutes of Health Grant No. R01-CA184173 and a grant from Varian Medical Systems.

TH-CD-303-06: Deformable Registration-Based Image Estimation Method for 4D CBCT Using Region-Based PCA

Purpose: To investigate the feasibility of using region-based principal component analysis (PCA) for motion modeling to estimate on-board 4D-CBCT using prior information and limited angle projections for potential 4D target verification of lung radiotherapy. Methods: A technique for lung 4D-CBCT reconstruction has been previously developed using a deformation field map (DFM)-based strategy. In this method, each phase of the 4D-CBCT is generated by deforming a prior CT volume. The DFM is solved by a motion modeling based on global PCA and free-form deformation (MM-FD) technique, using data fidelity constraint and the deformation energy minimization. For this study, a new region-based PCA method was developed to optimize the motion-modeling part of the MM-FD technique in order to estimate target delineation more accurately using fewer on-board projection angles. First, the DFMs are estimated using the region-based PCA divides the motion model extracted from planning 4D-CT into two regions: tumor and everything else. The motion model parameters of both regions are optimized together based on data fidelity constraint with relative weighting added to the tumor PCA model. The 4D digital extended cardiac-torso phantom was used to evaluate the region-based PCA MM-FD technique. A lung patient with a 30 mm diameter lesion was simulated with various anatomical and respirational changes from planning 4D CT to onboard volume. The reconstruction accuracy was evaluated by the Volume-Percent-Difference(VPD)/Center-of-Mass-Shift(COMS) between lesions in the estimated and “ground-truth” on board 4D CBCT. Results: For patient scenarios with various shifts of tumor from planning CT to on-board acquisition with 15 degree orthogonal projections, the average VPD/COMS for the MM-FD technique was 5.04%/0.26 mm and 23.23%/2.4mm for region-based and global PCA MM-FD, respectively. Conclusion: The region-based PCA MM-FD technique can potentially further reduces the scan angle needed for onboard 4D CBCT reconstruction for ultra-fast 4D verification. National Institutes of Health Grant No. R01-CA184173 Varian Medical System

Design-space dimensionality reduction in shape optimization by Karhunen–Loève expansion

The paper presents a methodology to reduce the dimension of design spaces in shape optimization problems, while retaining a desired level of geometric variance. The method is based on a generalized Karhunen–Loève expansion (KLE). Arbitrary shape modification spaces are assessed in terms of Karhunen–Loève modes (eigenvectors) and associated geometric variance (eigenvalues). The former are used as a basis in order to build a reduced-dimensionality representation of the shape modification. The method is demonstrated for the shape optimization of a high-speed catamaran, based on CFD simulations and aimed at the reduction of the wave component of calm-water resistance. KLE is applied to three design spaces with large dimensionality (≥20), based on a free form deformation technique. The space with the largest geometric variance is selected for dimensionality reduction and design optimization. N-dimensional design spaces are used, with N=1, 2, 3, and 4, retaining up to the 95% of the geometric variance associated to the original space. The correlation between the objective reduction achieved, the dimension N and the geometric variance of the reduced-dimensionality space is shown and found significant.

Image registration guided, sparsity constrained reconstructions for dynamic MRI

It is generally a challenging task to reconstruct dynamic magnetic resonance (MR) images with high spatial and high temporal resolutions, especially with highly incomplete k-space sampling. In this work, a novel method that combines a non-rigid image registration technique with sparsity-constrained image reconstruction is introduced. Employing a multi-resolution free-form deformation technique with B-spline interpolations, the non-rigid image registration accurately models the complex deformations of the physiological dynamics, and provides artifact-suppressed high spatial-resolution predictions. Based on these prediction images, the sparsity-constrained data fidelity-enforced image reconstruction further improves the reconstruction accuracy. When compared with the k-t FOCUSS with motion estimation/motion compensation (MEMC) technique on volunteer scans, the proposed method consistently outperforms in both the spatial and the temporal accuracy with variously accelerated k-space sampling. High fidelity reconstructions for dynamic systolic phases with reduction factor of 10 and cardiac perfusion series with reduction factor of 3 are presented.

SU-E-J-246: A Deformation-Field Map Based Liver 4D CBCT Reconstruction Method Using Gold Nanoparticles as Constraints

Purpose: To investigate the feasibility of using nanoparticle markers to validate liver tumor motion together with a deformation field map-based four dimensional (4D) cone-beam computed tomography (CBCT) reconstruction method. Methods: A technique for lung 4D-CBCT reconstruction has been previously developed using a deformation field map (DFM)-based strategy. In this method, each phase of the 4D-CBCT is considered as a deformation of a prior CT volume. The DFM is solved by a motion modeling and free-form deformation (MM-FD) technique, using a data fidelity constraint and the deformation energy minimization. For liver imaging, there is low contrast of a liver tumor in on-board projections. A validation of liver tumor motion using implanted gold nanoparticles, along with the MM-FD deformation technique is implemented to reconstruct onboard 4D CBCT liver radiotherapy images. These nanoparticles were placed around the liver tumor to reflect the tumor positions in both CT simulation and on-board image acquisition. When reconstructing each phase of the 4D-CBCT, the migrations of the gold nanoparticles act as a constraint to regularize the deformation field, along with the data fidelity and the energy minimization constraints. In this study, multiple tumor diameters and positions were simulated within the liver for on-board 4D-CBCT imaging. The on-board 4D-CBCT reconstructed by the proposed method was compared with the “ground truth” image. Results: The preliminary data, which uses reconstruction for lung radiotherapy suggests that the advanced reconstruction algorithm including the gold nanoparticle constraint will Resultin volume percentage differences (VPD) between lesions in reconstructed images by MM-FD and “ground truth” on-board images of 11.5% (± 9.4%) and a center of mass shift of 1.3 mm (± 1.3 mm) for liver radiotherapy. Conclusion: The advanced MM-FD technique enforcing the additional constraints from gold nanoparticles, results in improved accuracy for reconstructing on-board 4D-CBCT of liver tumor. Varian medical systems research grant

A new patient-specific planning method based on joint contact force balance with soft tissue release in total knee arthroplasty

Although the conventional total knee arthoplasty (TKA) planning used 2D or 3D images of the patient’s lower extremity to determine bone alignment, it was shown that good bone alignment does not always guarantee successful results due to unbalanced joint contact forces and soft tissue tension. Thus, the joint contact force balance and soft tissue tension were pre-operatively calculated using finite element analysis (FEA). Although the results of FEA are accurate, the rigid body spring method (RBSM) has introduced for the joint contact force or pressure distribution due to the high costs for FEA. In this paper, an innovative method was described to quantify the pre- and post-operative knee contact force balance using various engineering techniques, including the free-form deformation (FFD) technique, and RBSM-based cartilage contact force and ligament tension estimation algorithm. Nine cases of TKA planning were performed with three different surgical options, including over-correction of the valgus ratio of the femur, tibial plateau cutting length, and MCL release to redistribute the medial and lateral contact force balance. This method is useful as a patient-specific preoperative planning tool in TKA surgery.

A technique for estimating 4D‐CBCT using prior knowledge and limited‐angle projections

To develop a technique to estimate onboard 4D-CBCT using prior information and limited-angle projections for potential 4D target verification of lung radiotherapy. Each phase of onboard 4D-CBCT is considered as a deformation from one selected phase (prior volume) of the planning 4D-CT. The deformation field maps (DFMs) are solved using a motion modeling and free-form deformation (MM-FD) technique. In the MM-FD technique, the DFMs are estimated using a motion model which is extracted from planning 4D-CT based on principal component analysis (PCA). The motion model parameters are optimized by matching the digitally reconstructed radiographs of the deformed volumes to the limited-angle onboard projections (data fidelity constraint). Afterward, the estimated DFMs are fine-tuned using a FD model based on data fidelity constraint and deformation energy minimization. The 4D digital extended-cardiac-torso phantom was used to evaluate the MM-FD technique. A lung patient with a 30 mm diameter lesion was simulated with various anatomical and respirational changes from planning 4D-CT to onboard volume, including changes of respiration amplitude, lesion size and lesion average-position, and phase shift between lesion and body respiratory cycle. The lesions were contoured in both the estimated and "ground-truth" onboard 4D-CBCT for comparison. 3D volume percentage-difference (VPD) and center-of-mass shift (COMS) were calculated to evaluate the estimation accuracy of three techniques: MM-FD, MM-only, and FD-only. Different onboard projection acquisition scenarios and projection noise levels were simulated to investigate their effects on the estimation accuracy. For all simulated patient and projection acquisition scenarios, the mean VPD (±S.D.)∕COMS (±S.D.) between lesions in prior images and "ground-truth" onboard images were 136.11% (±42.76%)∕15.5 mm (±3.9 mm). Using orthogonal-view 15°-each scan angle, the mean VPD∕COMS between the lesion in estimated and "ground-truth" onboard images for MM-only, FD-only, and MM-FD techniques were 60.10% (±27.17%)∕4.9 mm (±3.0 mm), 96.07% (±31.48%)∕12.1 mm (±3.9 mm) and 11.45% (±9.37%)∕1.3 mm (±1.3 mm), respectively. For orthogonal-view 30°-each scan angle, the corresponding results were 59.16% (±26.66%)∕4.9 mm (±3.0 mm), 75.98% (±27.21%)∕9.9 mm (±4.0 mm), and 5.22% (±2.12%)∕0.5 mm (±0.4 mm). For single-view scan angles of 3°, 30°, and 60°, the results for MM-FD technique were 32.77% (±17.87%)∕3.2 mm (±2.2 mm), 24.57% (±18.18%)∕2.9 mm (±2.0 mm), and 10.48% (±9.50%)∕1.1 mm (±1.3 mm), respectively. For projection angular-sampling-intervals of 0.6°, 1.2°, and 2.5° with the orthogonal-view 30°-each scan angle, the MM-FD technique generated similar VPD (maximum deviation 2.91%) and COMS (maximum deviation 0.6 mm), while sparser sampling yielded larger VPD∕COMS. With equal number of projections, the estimation results using scattered 360° scan angle were slightly better than those using orthogonal-view 30°-each scan angle. The estimation accuracy of MM-FD technique declined as noise level increased. The MM-FD technique substantially improves the estimation accuracy for onboard 4D-CBCT using prior planning 4D-CT and limited-angle projections, compared to the MM-only and FD-only techniques. It can potentially be used for the inter/intrafractional 4D-localization verification.

Method to Geometrically Personalize a Detailed Finite-Element Model of the Spine

To date, developing geometrically personalized and detailed solid finite-element models (FEMs) of the spine remains a challenge, notably due to multiple articulations and complex geometries. To answer this problem, a methodology based on a free-form deformation technique (kriging) was developed to deform a detailed reference finite-element mesh of the spine (including discs and ligaments) to the patient-specific geometry of 10- and 82-year-old asymptomatic spines. Different kriging configurations were tested: with or without smoothing, and control points on or surrounding the entire mesh. Based on the results, it is recommended to use surrounding control points and smoothing. The mean node to surface distance between the deformed and target geometries was 0.3±1.1 mm. Most elements met the mesh quality criteria (95%) after deformation, without interference at the articular facets. The method's novelty lies in the deformation of the entire spine at once, as opposed to deforming each vertebra separately, with surrounding control points and smoothing. This enables the transformation of reference vertebrae and soft tissues to obtain complete and personalized FEMs of the spine with minimal postprocessing to optimize the mesh.

Hermite approximation for free-form deformation of curves and surfaces

Free-Form Deformation Techniques (FFD) are commonly used to generate animations, where a polygonal approximation of the final object suffices for visualization purposes. However, for some CAD/CAM applications, we need an explicit expression of the object, rather than a collection of sampled points. If both object and deformation are polynomial, their composition yields a result that is also polynomial, albeit very high degree, something undesirable in real applications. To solve this problem, we transform each curve or surface composing the object, usually expressed in the Bernstein basis, to a modified Newton form. In this representation, the two-point analogue of Taylor expansions, the composition admits a simple expression in terms of discrete convolutions, and degree reduction corresponding to Hermite approximation is trivial by dropping high-degree coefficients. Furthermore, degree-reduction can be incorporated into the composition. Finally, the deformed curve or surface is converted back to the Bernstein form. This method extends to general non-polynomial deformation, such as bending and twisting, by computing a polynomial approximant of the deformation.