**3. Results**

Figure 6 shows the 2D spatial distribution map and corresponding error map for each mechanical property in each simulation test. Table 1 shows the percentage of the region in the 2D spatial distribution map consisting of elements with a simulation value within ±10% of the theoretical value for each simulation test.

*Materials* **2021**, *14*, x FOR PEER REVIEW 12 of 20

error larger than 10%.

than 10%.

In the third simulation test (when the uniform compressional pressure is exerted on the entire top surface, and the bottom is fixed along the vertical direction while the side is fixed along the horizontal direction), the 2D distribution map for each mechanical property is perfectly homogeneous. However, only ߬ோ can be accurately measured, while ܧ and ݃ cannot be since each element in the entire region has an error larger than 10%.

In the fourth and fifth simulation tests (when the uniform compressional pressure is exerted on the half of the top surface, and the bottom is fixed along the vertical direction), ݃ and ߬ோ can be accurately measured, except for a small region close to the top and/or bottom. ܧ cannot be accurately measured since each element in the entire region has an

In the sixth simulation test (when the uniform compressional pressure is exerted on the half of the top surface, and the bottom is fixed along the vertical direction while the side is fixed along the horizontal direction), ݃ can be accurately measured in most of the region (96.18%). For ߬ோ, it can only be accurately measured in 23.24% of the region. ܧ cannot be accurately measured since each element in the entire region has an error larger

**Figure 6.** The 2D spatial distribution and corresponding error maps. In the error map, the yellow color means the error is larger than 10%. (**a**) Simulation test 1. (**b**) Simulation test 2. (**c**) Simulation test 3. (**d**) Simulation test 4. (**e**) Simulation test 5. (**f**) Simulation test 6. **Figure 6.** The 2D spatial distribution and corresponding error maps. In the error map, the yellow color means the error is larger than 10%. (**a**) Simulation test 1. (**b**) Simulation test 2. (**c**) Simulation test 3. (**d**) Simulation test 4. (**e**) Simulation test 5. (**f**) Simulation test 6.

**Table 1.** The percentage of the region in the map consisting of elements with the simulation value within ±10% of the theoretical value set in ABAQUS. The larger this percentage, the more accurate the measurement. **Table 1.** The percentage of the region in the map consisting of elements with the simulation value within ±10% of the theoretical value set in ABAQUS. The larger this percentage, the more accurate the measurement.


The findings of the present finite element analysis study provide a reference for experimental phantom designs regarding loading and boundary conditions, as well as guidance towards validating the experimental results of a compressional viscoelastography system, such as the one in the literature [33,34]. The most important finding is that the accuracy of compressional viscoelastography for measuring the viscoelastic properties of materials is only excellent if the compressional pressure is exerted on the entire top surface of the sample, as well as if the bottom of the sample is fixed just along the vertical direction. These findings imply that in an experimental validation study, the phantom design should take into account that the surface area of the pressure plate must be equal to or larger than that of the top surface of the sample, and the sample should be placed directly on the testing platform without any fixation (such as a sample container). However, in an experimental design, the sample may slip horizontally under loading if there is no fixation to stabilize the sample. Fortunately, the slip can be prevented by using a pressure plate

with a surface area significantly larger than that of the top surface of the sample.

In the present study, it was found that the loading and boundary conditions in computational simulations of compressional viscoelastography severely affect the accuracy of measuring the modulus of elasticity, ܧ. If the compressional pressure is exerted on half of the top surface of the sample and the boundary condition is more complex than the abovementioned optimal condition (the bottom of the sample is fixed just along the vertical direction), ܧ is difficult measure accurately. This implies that if the area of the pressure plate is smaller than that of the top surface of the sample, and if the sample is secured by any external fixation method (such as the sample container shown in Figure 2b), ܧ may not be accurately measured. In these kinds of conditions, there will be horizontal strains occurring within the sample. The theory used in the present study for analyzing the creep

**4. Discussion** 

In the first simulation test (when the uniform compressional pressure is exerted on the entire top surface, and the bottom is fixed along the vertical direction while the side is not fixed), the 2D distribution map for each mechanical property is perfectly homogeneous, as is the corresponding error map. The error value at each element in the error map is nearly zero. This means that all three mechanical properties can be accurately measured in this case.

In the second simulation test (when the uniform compressional pressure is exerted on the entire top surface, and the bottom is fixed along all directions while the side is not fixed), *g* can be accurately measured in most of the region (92.42%), except for a small region close to the bottom. *τ<sup>R</sup>* can be accurately measured in 77.5% of the entire region, except for a region extending from the bottom to near the depth of 40 mm. For *E*, it can only be accurately measured in 37.28% of the entire region, between the depth of 5 to 25 mm approximately, and there is a significant region where each element in this region has an error larger than 10%.

In the third simulation test (when the uniform compressional pressure is exerted on the entire top surface, and the bottom is fixed along the vertical direction while the side is fixed along the horizontal direction), the 2D distribution map for each mechanical property is perfectly homogeneous. However, only *τ<sup>R</sup>* can be accurately measured, while *E* and *g* cannot be since each element in the entire region has an error larger than 10%.

In the fourth and fifth simulation tests (when the uniform compressional pressure is exerted on the half of the top surface, and the bottom is fixed along the vertical direction), *g* and *τ<sup>R</sup>* can be accurately measured, except for a small region close to the top and/or bottom. *E* cannot be accurately measured since each element in the entire region has an error larger than 10%.

In the sixth simulation test (when the uniform compressional pressure is exerted on the half of the top surface, and the bottom is fixed along the vertical direction while the side is fixed along the horizontal direction), *g* can be accurately measured in most of the region (96.18%). For *τR*, it can only be accurately measured in 23.24% of the region. *E* cannot be accurately measured since each element in the entire region has an error larger than 10%.
