Proportional Topology Optimization: A new non-gradient method for solving stress constrained and minimum compliance problems and its implementation in MATLAB

A new topology optimization method called the Proportional Topology Optimization (PTO) is presented. As a non-gradient method, PTO is simple to understand, easy to implement, and is also efficient and accurate at the same time. It is implemented into two MATLAB programs to solve the stress constrained and minimum compliance problems. Descriptions of the algorithm and computer programs are provided in detail. The method is applied to solve three numerical examples for both types of problems. The method shows comparable efficiency and accuracy with an existing gradient optimality criteria method. Also, the PTO stress constrained algorithm and minimum compliance algorithm are compared by feeding output from one algorithm to the other in an alternative manner, where the former yields lower maximum stress and volume fraction but higher compliance compared to the latter. Advantages and disadvantages of the proposed method and future works are discussed. The computer programs are self-contained and publicly shared in the website www.ptomethod.org.Comment: 18 pages, 8 figures, and 2 appendices (MATLAB codes

Multiresolution Molecular Mechanics: Dynamics, Adaptivity, and Implementation

Full atomistic Molecular Dynamics (MD) simulations are very accurate but too costly; however, atomistic resolution is not actually required everywhere in many problems. For this reason, a concurrent atomistic/continuum coupling method called Multiresolution Molecular Mechanics (MMM) has been developed. The method employs atomistic resolution in the localized regions of interest and coarser continuum description elsewhere. A number of such multiscale methods have been developed but they fail to demonstrate consistency, accuracy, adaptivity, flexibility, and efficiency all in one. The goal of this research is thus to develop a multiscale method that possesses these properties to outperform the MD method by 1) formulating new dynamics equations under the MMM framework, 2) developing an adaptivity scheme, and 3) implementing efficient algorithms for the method. First, the derivation of the governing MMM equations from a Hamiltonian that approximates the energy of the original system is presented. Second, the adaptivity analysis of the MMM method is presented. Refinement and coarsening mechanisms of the adaptivity scheme are described in detail and the step-by-step procedures are outlined. Third, the implementation and efficiency of the MMM software is presented. The structure of the software along with the associated technologies is introduced. Many improvements that contribute to the efficiency of the MMM software are described and demonstrated through benchmark tests. The efficiency of the software is found to be as good as one of the best state-of-the-art MD codes, i.e., LAMMPS. The speed-up of the code in proportion to reduction in the rep atom ratio is demonstrated. The scalability of the software is demonstrated and competing effects of multiscale modeling and parallelization is discussed. The dynamics, adaptivity, and efficiency of the method are demonstrated by numerical examples including wave and crack propagation, dislocation glide, nanoindentation, and modal analysis in 1/2/3 dimensions. All results agree well with the true full atomistic solutions. Ultimately, the MMM method demonstrates an improvement of 6.3 – 8.3 times in efficiency over MD method by means of a combined reduction in simulation time and number of processors. In conclusion, this dissertation shows that the MMM method is consistent, accurate, flexible, and efficient

Numerical examples: (a) MBB beam–only right half (a2) of the full design domain (a1) is considered due to symmetry, (b) Cantilever beam, and (c) L bracket.

<p>Numerical examples: (a) MBB beam–only right half (a2) of the full design domain (a1) is considered due to symmetry, (b) Cantilever beam, and (c) L bracket.</p

Comparison of stress versus volume fraction curves of PTOs and PTOc for the MBB beam (left), cantilever beam (center), and L bracket (right) examples.

<p>Dashed lines indicate the links between PTOs and PTOc. A horizontal dashed line means stress output of PTOc is input to the PTOs and a vertical dashed line means volume fraction output of PTOs is input to PTOc.</p

Parametric results for the PTOc: Compliances (iteration number) are given for <i>α</i> from 0 to 0.9 and <i>q</i> from 0.25 to 3.

<p>Parametric results for the PTOc: Compliances (iteration number) are given for <i>α</i> from 0 to 0.9 and <i>q</i> from 0.25 to 3.</p

Topologies and compliance (PTOc) or stress (PTOs) distributions obtained from the MBB beam, cantilever beam, and L bracket examples.

<p>Topologies and compliance (PTOc) or stress (PTOs) distributions obtained from the MBB beam, cantilever beam, and L bracket examples.</p

Biochemical bone markers in nephrotic children

In this study we evaluated the effects of high-dose corticosteroid (CS) therapy and the character of the nephrotic syndrome (NS) itself on bones in patients with normal glomerular filtration rate. We measured serum osteocalcin (OC), alkaline phosphatase (ALP), intact parathyroid hormone (iPTH), 25-hydroxyvitamin D, calcium (Ca), phosphorus (P), and magnesium (Mg) levels, and urinary Ca and protein excretion in nephrotic children during the active phase before (group Ia) and after CS treatment (group Ib). The results were compared with age-matched control subjects. A significant increase in urinary Ca excretion was observed after CS treatment. Serum ALP, OC, and iPTH levels were within normal limits at the time of study entry. However, both serum OC and ALP levels showed a significant decrease after the completion of CS treatment (OC from 13.6+/-9.2 ng/ml to 6.7+/-5.2 ng/ml and ALP from 151.8+/-60.2 U/l to 116+/-43.8 U/l). 25-Hydroxyvitamin D levels increased to 17.2+/-8.9 mug/l from 9.9+/-6.9 mug/l after CS treatment. The effects of recurrent use of CSs were assessed by dividing nephrotic patients into two subgroups: infrequent relapsers (IFR) and frequent relapsers (FR). The cumulative dose of CS was 28,125 mg/m(2) for IFR and 105,000 mg/m(2) for FR. The changes in OC, ALP, and 25-hydroxyvitamin D levels after CS treatment were significantly different between IFR and FR. We conclude that high-dose CS treatment causes a decrease in bone formation, as shown by the changes in OC and ALP levels. 25-Hydroxyvitamin D levels remained lower than control subjects after CS therapy. The higher the cumulative dose of CS used the more marked the changes in biochemical bone markers. The contribution of FR to baseline 25-hydroxyvitamin D levels needs further study

Comparison of compliance versus volume fraction curves of PTOc and Top88 for the MBB beam (left), cantilever beam (center), and L bracket (right) examples.

<p>Comparison of compliance versus volume fraction curves of PTOc and Top88 for the MBB beam (left), cantilever beam (center), and L bracket (right) examples.</p

Comparison of topologies of PTOc and Top88 for the MBB beam, cantilever beam, and L bracket examples.

<p>Comparison of topologies of PTOc and Top88 for the MBB beam, cantilever beam, and L bracket examples.</p

Comparison of iteration numbers and simulation times of Top88 and PTOc.

<p>Comparison of iteration numbers and simulation times of Top88 and PTOc.</p