How to resolve SharePoint 2013 server failed to communicate with SQL Server?

it might be caused by that the account you used to log into the server which didn’t have full access to SQL server. an account must have sysadmin, dbcreator and securityadmin rights

is share point and sql are on the same domain?
have you create a dns entry
you may have to log off and log back in to reflect the changes

@Tahir, it was working before, everything's fine and running smoothly, until we changed the IP of our whole system, then mess happened...

@Walter, please see the attached screenshot from the result of our test connection (SP -> SQL Server)

Then what shall we do next?
Screen-Shot-2018-04-12-at-11.26.31-A.png

Check the DNS
And
Firewall  (was it open on a specific IP address )

Follow up to our comment with Walter, we did that test connection with our MOSS 2007, the result is same, "Test connection succeeded".

@Tahir, yesterday, we did that already, first with our MOSS 2007, check the DNS, and even the hosts file, we checked everything and changed the IP addresses, that's why from our MOSS 2007, we run the SP Config successfully, we can open the CA, but the site that we use everyday, some parts are gone, and from CA, the SSA showed error (no connection from SQL). Anyway, please see our other thread regarding to this MOSS 2007.
And with our SP2013, we can't connect with our SQL DB, but when we test the connection, it succeeded...

@Walter, after we test the connection, it has a connection to our SQL, then we test our CA, but it resulted 500 Internal Server Error. Can't access the page...

Could it be an incorrect alias?
Open SQL Server Client Network Utility by typing cliconfg.exe in your Run command.
Check if there are any aliases defined for the server you are trying to connect to.
If present, do the following:
*    Click on Edit and rename the Server alias. (for example, if your server name is MySQL, rename it as MySQL_test) and retry the connection. If the connection works, it is an indication that you had an incorrect alias, probably from an old configuration that is no longer needed. If you continue to experience the error, rename the alias back to its original name and proceed to the next step.
*    Check the Connection parameters for the alias and ensure they are correct. Some of the common scenarios that can cause problems with connectivity are:
     *   Incorrect IP address for the Server name parameter. Ensure this IP address matches the entry in the SQL ErrorLog file.
     *   Incorrect Server name in the Server name parameter- For example though your Server alias points the correct server name, if the Server name parameter has incorrect value, connections will fail.
      *  If you are using a named pipes alias, ensure the pipe name has correct format
         *   For connecting to default instance named Mydefaultinstance, the pipe name should be \\Mydefaultinstance\pipe\sql\query
         *   For connecting to a named instance MySQL\Named, the pipe name should be \\MySQL\pipe\MSSQL$Named\sql\query

Hi Mike! Thanks for your response...
>>Open SQL Server Client Network Utility by typing cliconfg.exe in your Run command.
Check if there are any aliases defined for the server you are trying to connect to.<<
- We did that both sides, from our SQL Server and even in our SP 2013 server, renamed the Alias / Server name or return back to its original name, the error still persists...

Kindly see the attached SQL Error log, it's surprising that the 2 admin accounts can't be use to access the db, but receive error...
SQL_Error.log

Hope you have this solved by now. Some very good advice has been given by all. Here is one of your issues:

@Walter, after we test the connection, it has a connection to our SQL, then we test our CA, but it resulted 500 Internal Server Error. Can't access the page..

Since there is a connection to SQL, this may be caused by IIS on the web server. Check to make sure that the app pool is running, or better yet recycle the app pool in IIS. If you still have an error do an IIS reset. This error could be caused the SQL services on the SQL server being stopped. That can be checked on the SQL server directly.

Good luck...

@Walter, thank you for your kind response...
>> Since there is a connection to SQL, this may be caused by IIS on the web server. Check to make sure that the app pool is running, or better yet recycle the app pool in IIS.<<
- Done recycling...

>> If you still have an error do an IIS reset.<<
- Done IISRESET

>> This error could be caused the SQL services on the SQL server being stopped.<<
- Checked the SQL services on SQL server, restarted all related to SQL.

Result: Still 500 Internal Server Error...

Please see the attached screenshots...
Screen-Shot-2018-04-17-at-10.42.08-P.png
Screen-Shot-2018-04-17-at-10.44.22-P.png

Make sure the credentials for the app pools are set correctly. You should do this under normal conditions via Central Administration (does that work?), but in this recovery state you can do it directly in IIS. What you need to do is right click on the app pool named SharePoint - 2018. then you will see a button for advanced settings, and from there you can add the user name and password. Give that a try.

Questions, why is that named SharePoint 2018. I thought you installed your SharePoint farm last year or the year before.

Good luck...

CA is still not working... So attaching the screenshot from the Advance Settings of SharePoint - 2018 app pool.
Where to put the user name and password?

Q: Questions, why is that named SharePoint 2018. I thought you installed your SharePoint farm last year or the year before.
A: It's because we started to setup SP 2013 since last year 2015, then we tested some stuffs from virtual player, and we've used several ports already until we come up with SharePoint - 2018 (our latest one)...
Screen-Shot-2018-04-17-at-11.47.06-P.png
Screen-Shot-2018-04-17-at-11.47.29-P.png

Look at the first screen shot. About half way down is a section titled "Process Model". The first entry in the section is "Identity". Click on the entry and it should open so that you can reenter the username and password.

I understand about where the name came from.

Though we're not yet done with this, we would like to reward points to the closest and give us idea in some parts... Thank you!

Thanks