Alok Sutradhar

Transient heat conduction in homogeneous and non-homogeneous materials by the Laplace transform Galerkin boundary element method

The Greens function for three-dimensional transient heat conduction (diffusion equation) for functionally graded materials (FGMs) is derived. The thermal conductivity and heat capacitance both vary exponentially in one coordinate. In the process of solving this diffusion problem numerically, a Laplace transform (LT) approach is used to eliminate the dependence on time. The fundamental solution in Laplace space is derived and the boundary integral equation formulation for the Laplace Transform boundary element method (LTBEM) is obtained. The numerical implementation is performed using a Galerkin approximation, and the time-dependence is restored by numerical inversion of the LT. Two numerical inversion techniques have been investigated: a Fourier series method and Stehfests algorithm, the latter being preferred. A number of test problems have been examined, and the results are in excellent agreement with available analytical solutions.

Recent Advances and Emerging Applications of the Boundary Element Method

Sponsored by the U.S. National Science Foundation, a workshop on the boundary element method (BEM) was held on the campus of the University of Akron during September 1–3, 2010 (NSF, 2010, “Workshop on the Emerging Applications and Future Directions of the Boundary Element Method,” University of Akron, Ohio, September 1–3). This paper was prepared after this workshop by the organizers and participants based on the presentations and discussions at the workshop. The paper aims to review the major research achievements in the last decade, the current status, and the future directions of the BEM in the next decade. The review starts with a brief introduction to the BEM. Then, new developments in Greens functions, symmetric Galerkin formulations, boundary meshfree methods, and variationally based BEM formulations are reviewed. Next, fast solution methods for efficiently solving the BEM systems of equations, namely, the fast multipole method, the pre-corrected fast Fourier transformation method, and the adaptive cross approximation method are presented. Emerging applications of the BEM in solving microelectromechanical systems, composites, functionally graded materials, fracture mechanics, acoustic, elastic and electromagnetic waves, time-domain problems, and coupled methods are reviewed. Finally, future directions of the BEM as envisioned by the authors for the next five to ten years are discussed. This paper is intended for students, researchers, and engineers who are new in BEM research and wish to have an overview of the field. Technical details of the BEM and related approaches discussed in the review can be found in the Reference section with more than 400 papers cited in this review.

Topological optimization for designing patient-specific large craniofacial segmental bone replacements.

Restoring normal function and appearance after massive facial injuries with bone loss is an important unsolved problem in surgery. An important limitation of the current methods is heuristic ad hoc design of bone replacements by the operating surgeon at the time of surgery. This problem might be addressed by incorporating a computational method known as topological optimization into routine surgical planning. We tested the feasibility of using a multiresolution three-dimensional topological optimization to design replacements for massive midface injuries with bone loss. The final solution to meet functional requirements may be shaped differently than the natural human bone but be optimized for functional needs sufficient to support full restoration using a combination of soft tissue repair and synthetic prosthetics. Topological optimization for designing facial bone tissue replacements might improve current clinical methods and provide essential enabling technology to translate generic bone tissue engineering methods into specific solutions for individual patients.

Experimental validation of 3D printed patient-specific implants using digital image correlation and finite element analysis

With the dawn of 3D printing technology, patient-specific implant designs are set to have a paradigm shift. A topology optimization method in designing patient-specific craniofacial implants has been developed to ensure adequate load transfer mechanism and restore the form and function of the mid-face. Patient-specific finite element models are used to design these implants and to validate whether they are viable for physiological loading such as mastication. Validation of these topology optimized finite element models using mechanical testing is a critical step. Instead of inserting the implants into a cadaver or patient, we embed the implants into the computer-aided skull model of a patient and, fuse them together to 3D print the complete skull model with the implant. Masticatory forces are applied in the molar region to simulate chewing and measure the stress-strain trajectory. Until recently, strain gages have been used to measure strains for validation. Digital Image Correlation (DIC) method is a relatively new technique for full-field strain measurement which provides a continuous deformation field data. The main objective of this study is to validate the finite element model of patient-specific craniofacial implants against the strain data from the DIC obtained during the mastication simulation and show that the optimized shapes provide adequate load-transfer mechanism. Patient-specific models are obtained from CT scans. The principal maximum and minimum strains are compared. The computational and experimental approach to designing patient-specific implants proved to be a viable technique for mid-face craniofacial reconstruction.

Designing patient-specific 3D printed craniofacial implants using a novel topology optimization method.

Large craniofacial defects require efficient bone replacements which should not only provide good aesthetics but also possess stable structural function. The proposed work uses a novel multiresolution topology optimization method to achieve the task. Using a compliance minimization objective, patient-specific bone replacement shapes can be designed for different clinical cases that ensure revival of efficient load transfer mechanisms in the mid-face. In this work, four clinical cases are introduced and their respective patient-specific designs are obtained using the proposed method. The optimized designs are then virtually inserted into the defect to visually inspect the viability of the design . Further, once the design is verified by the reconstructive surgeon, prototypes are fabricated using a 3D printer for validation. The robustness of the designs are mechanically tested by subjecting them to a physiological loading condition which mimics the masticatory activity. The full-field strain result through 3D image correlation and the finite element analysis implies that the solution can survive the maximum mastication of 120 lb. Also, the designs have the potential to restore the buttress system and provide the structural integrity. Using the topology optimization framework in designing the bone replacement shapes would deliver surgeons new alternatives for rather complicated mid-face reconstruction.

In vivo measurement of breast skin elasticity and breast skin thickness

The mechanical properties of the breast skin play an important role in explaining the changes associated with radiotherapy, tissue expansion, and breast reconstruction surgery. Quantitative measurement of mechanical properties of breast skin is essential for surgical preplanning and outcome prediction. We have measured the skin elasticity properties and skin thickness of the breast using noninvasive methods.

Boundary element methods for functionally graded materials

Functionally graded materials (FGMs) possess a smooth variation of material properties due to continuous change in microstructural details. For example, the material gradation may change gradually from a pure ceramic to a pure metal. This work focuses on potential (both steady state and transient) and elasticity problems for inhomogeneous materials. The Green’s function(GF) for these materials (e.g. exponentially graded) are expressed as the GF for the homogeneous material plus additional terms due to material gradation. The numerical implementations are performed using a Galerkin (rather than collocation) approximation. A number of examples have been carried out. The results of some speciflc test problems agree within plotting accuracy with available analytical solutions.

Design of complex bone internal structure using topology optimization with perimeter control

Large facial bone loss usually requires patient-specific bone implants to restore the structural integrity and functionality that also affects the appearance of each patient. Titanium alloys (e.g., Ti-6Al-4V) are typically used in the interfacial porous coatings between the implant and the surrounding bone to promote stability. There exists a property mismatch between the two that in general leads to complications such as stress-shielding. This biomechanical discrepancy is a hurdle in the design of bone replacements. To alleviate the mismatch, the internal structure of the bone replacements should match that of the bone. Topology optimization has proven to be a good technique for designing bone replacements. However, the complex internal structure of the bone is difficult to mimic using conventional topology optimization methods without additional restrictions. In this work, the complex bone internal structure is recovered using a perimeter control based topology optimization approach. By restricting the solution space by means of the perimeter, the intricate design complexity of bones can be achieved. Three different bone regions with well-known physiological loadings are selected to illustrate the method. Additionally, we found that the target perimeter value and the pattern of the initial distribution play a vital role in obtaining the natural curvatures in the bone internal structures as well as avoiding excessive island patterns.

Automated cross-sectional shape recovery of 3D branching structures from point cloud

Abstract Many applications rely on scanned data, which can come from a variety of sources: optical scanners, coordinate measuring machines, or medical imaging. We assume that the data input to these applications is an unorganized point cloud or mesh of vertices. The objective may be to find particular features (medical diagnostics or reverse engineering) or comparison to some reference geometry (e.g. dimensional metrology). This paper focuses on the feature fitting of a segmented point cloud, specifically for branched, organic structures or structural frames, and targets non-monolithic geometries. In this paper, a methodology is presented for the automated recovery of cross-sectional shapes using centrally located curves. We assume a triangulated surface mesh is generated from the scanned point cloud. This surface mesh is the starting point for our methodology. We then find the curve skeleton of the part which abstractly describes the global geometry and topology. Next after segmenting the curve skeleton into non-branching segments, orthogonal planes to the curve skeleton segments, at preset or adaptive intervals, make slices through the surface mesh edges. The intersection points are extracted creating a 2D point cloud of the cross section. A number of application specific post-processing operations can be performed after obtaining the 2D point cloud cross sections and the curve skeleton paths including: calculations such as area or area moments of inertia, feature fitting or recognition, and digital shape reconstruction. Case studies are presented to demonstrate capabilities and limitations, and to provide insight into appropriate uses and adaptations for the presented methodology.

Bean: Boundary Element Analysis Program

Synopsis: This chapter is devoted to a description of the educational MATLAB computer code Boundary Element ANalysis, including its graphical user interface. BEAN is a user-friendly adaptive symmetric Galerkin BEM code to solve the twodimensional Laplace Equation. This chapter outlines both the specific procedures to set up 2-D Laplace problems and the steps to utilize BEAN’s post-processing capabilities. The book web-site contains additional related material i.e. example library and a tutorial video.