Optimization of SLM Process Parameters for Ti6Al4V Medical Implants
Summary (3 min read)
1. INTRODUCTION
- Process-induced variability has huge impacts on the circuit performance in the sub-90nm VLSI technologies [10, 9].
- One important aspect of the variations comes from the chip leakage currents.
- Io f f and threshold voltage Vth as shown below [14], Io f f = Is0e Vgs−Vth nVT (1− e −Vds VT ) (1) where Is0 is a constant related to the device characteristics, VT is the thermal voltage, and n is a constant.
- But this method needs to know the impulse response from all the current sources to all the nodes, which is expensive to compute for a large network.
2. PROBLEM FORMULATION
- The authors first present the model of power grids in this paper.
- The authors then present the modeling issue of leakage current under intra-die variations.
- After this, the authors present the problem that they try to solve.
2.1 Power Grid Network Models
- The power grid networks in this paper are modeled as RC networks with known time-variant current sources, which are obtained by gate level logic simulations of the VLSI systems.
- For a power grid (versus the ground grid), some nodes have known voltage modeled as constant voltage sources.
- For C4 power grids, the known voltage nodes can be internal nodes inside the power grid.
- Given known deterministic vector of current sources, I(t), the node voltages can be obtained by solving the following differential equations, which is formulated using modified nodal analysis (MNA) approach, Gv(t)+C dv(t) dt = I(t) (2) where G is the conductance matrix, C is admittance matrix resulting from capacitive elements.
- V(t) is the vector of time-varying node voltages and branch currents of voltage sources that the authors try to solve.
2.2 Modeling Leakage Current Variations
- The G and C matrices and input currents I(t) depend on the circuit parameters, such as metal wire width, length, thickness on power grids, and transistor parameters, such as channel length, width, gate oxide thickness, etc.
- The authors only consider the log leakage current variation, due to the channel length variations, which is modeled as Gaussian variations [12].
- After that, the authors consider spatial correlation in the intra-die variation.
- Therefore, given the process variations, the MNA for (2) becomes Gv(t)+C dv(t) dt = I(t,ξ(θ)) (3) Note that the input current vector, I(t,ξ(θ)), has both deterministic and random components.
3.1 Concept of Hermite Polynomial Chaos
- In the following, a random variable ξ(θ) is expressed as a function of θ, which is the random event.
- Hermite PC utilizes a series of orthogonal polynomials (with respect to the Gaussian distribution) to facilitate stochastic analysis [16].
- These polynomials are used as the orthogonal basis to decompose a random process in a similar way that sine and cosine functions are used to decompose a periodic signal in Fourier series expansion.
3.2 Simulation Approach Based on Hermite PCs
- In case that v(t,ξ) is unknown random variable vector (with unknown distributions) like node voltages in (3), then the coefficients can be computed by using Galerkin method, which states that the best approximation of v(t,ξ) is obtained when the error ∆(t,ξ), which is defined as ∆(t,ξ) = Gv(t)+C dv(t) dt − I(t,ξ(θ)) (10) is orthogonal to the approximation.
- Once the authors obtain those coefficients, the mean and variance of the random variables can be easily computed as shown later in the section.
- This will be explained in details in the next section.
4. LOG-NORMAL LEAKAGE CURRENT VARIATIONS
- The authors present the new method for representing the lognormal leakage current distributions by using Hermite PCs with one or more independent Gaussian variables representing the channel length or threshold voltage variations.
- The authors method is based on [6] and the authors will show how it can be applied to solve their problems for one or more independent Gaussian variables.
4.1 Hermite Polynomial Chaos representation
- Let g(ξ) be the Gaussian random variable, denoting threshold voltage or device channel length.
- (18) Obviously, for the MOS device leakage current equation (1), leakage current, Io f f = cIl(Vth) = ce −Vth , where the leakage component Il(Vth) is a log-normal random variable.
- To find the other coefficients, the authors can apply (9) on l(ξ).
4.3 Hermite PC with two and more Gaussian variables
- Similarly, for four Gaussian random variables, assume that ξ = [ξ1,ξ2,ξ3,ξ4] is a normalized, uncorrelated Gaussian random variable vector.
- Hence, the desired Hermite PC coefficients can be expressed using the equation (30) above.
5. VARIATIONS IN WIRES AND LEAKAGE CURRENTS
- The authors will consider variations in width (W ), thickness(T ) of wires of power grids, as well as threshold voltage(Vth) in active devices which are reflected in the leakage currents.
- The variation in width W and thickness T will cause variation in conductance matrix G and capactance matrix C while variation in threshold voltage will cause variation in leakage currents.
- As mentioned in previous section, the variation in leakage current resulting can be represented by a second Hermite PC as in equation (26): I(t,ξI) = I0(t)+ I1(t)ξI + I2(t)(ξ2I −1) (33) here, ξi is a normalized Guassian distribution random variable representing variation in threshold voltage.
6. EXPERIMENTAL RESULTS
- This section describes the simulation results of circuits with lognormal leakage current distributions for a number of power grid networks.
- All the proposed methods have been implemented in Matlab.
- All the experimental results are carried out in a Linux system with dual Intel Xeon CPUs with 3.06Ghz and 1GB memory.
6.2 Experiment considering leakage current variation
- Fig.1 shows the node voltage distributions at one node of a ground network with 1720 nodes.
- The standard deviations of the log-normal current sources with one Gaussian variable is 0.1.
- To consider multiple random variables, the authors divide the circuit into several partitions.
- Table 2 shows the speedup of the Hermite PC method over Monte Carlo method with 3000 samples.
- Also, one observation is that the speedup depends on the sampling size in MC method.
6.3 Experiment considering variation in G,C,I
- Considering variation in conductance, capacitor and leakage current at the same time, Fig. 3 and fig.
- 4 show the node voltage distributions at one node of two different ground circuit, Circuit1 and Circuit2, respectively.
- Table 3 shows the CPU speedup of HPC method than MC method.
- The sample time of Monte Carlo is 3500 and the authors can see that it’s more than 100X time faster of proposed method than the Monte Carlo method.
- The advantage is obviously seen in the table.
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Frequently Asked Questions (12)
Q2. What were the measurements used to determine the quality of the fabricated samples?
Porosity content, surface roughness, elastic modulus and compressive strength (UCS) were measured as outputs to better understand the quality characteristics of the fabricated samples.
Q3. What is the effect of laser power on surface roughness?
In addition, the increased laser power increases the energy density which improves the wettability of the melt pool, eliminating the differences in surface tension and in turn decreasing the chance of encountering the balling phenomenon which dramatically decreases the side surface roughness [2].
Q4. What is the effect of increasing the scan speed and hatch spacing on the melt pool?
Increasing the scan speed and hatch spacing and/or a decrease in the laser power shall reduce the melt pool and lead to incomplete consolidation.
Q5. What is the effect of low scan speed on the overlapping area of adjacent lines?
Small hatching spacing would increase the overlapping area of adjacent scanning lines, resulting in a complete melting of the powderbetween scanning lines.
Q6. How many GPa did a structure with a porosity of 43 and 80?
During the manufacturing of Ti6Al4V open-porous scaffolds using SLM, Weißmann and co-authors [43] concluded that a structure with a porosity % between 43 and 80 experienced an elastic modulus in the range from 26.3 to 3.4 GPa and an UCS in the range from 750 to 100 MPa.
Q7. How many GPa would be predicted for a SLM part?
In the current study it was predicted that at 23.62% porosity the elastics modulus and UCS of the SLM part would be 30 GPa and 522 MPa, respectively.
Q8. What is the effect of stress shielding on the implant?
Stress shielding prevents the needed stress being transferred from the implant to adjacent bone, which might result in bone loss in the near-vicinity of implants.
Q9. What is the role of pores in the fabrication of titanium implants?
Their results showed that creating pores in a Ti6Al4V part had a significant role in reducing its stiffness, which could allow the implant to have an elastic modulus that is close to that of human cortical bone.
Q10. What is the way to make a Ti implant?
In biomedical applications, a Ti implant with structure similar to that in sample 2 is recommended as it has low elastic modulus.
Q11. What was the effect of decreasing the porosity % on the elastic modulus?
Decreasing the porosity % from 25.43 to 2.94 resulted in a significant increase in the elastic modulus from 17 to 75 GPa, and a comparable rise in the UCS from 388 to 1749 MPa.
Q12. What is the way to use the implants?
for biomedical application, implants with rough surfaces are preferred to allow tissues to grow inside and integrating them to the hosting bones.